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Log & Exponential Function Properties

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Wize University Calculus 1 Textbook > Applications of Differentiation

Intervals of Increase and Decrease

3 Activities

Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Logarithmic Functions

4 Activities

Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Exponential Functions

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Which of the following functions is/are always increasing?
More Intervals of Increase and Decrease Questions:
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
Practice: Derivatives as a Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Intervals of Increase and Decrease
If the function is given by f(x)=∫x2(t+1)2(t−2)t dtf(x)=\int_x^2\left(t+1\right)^2\left(t-2\right)t\ dt, then f(x)f\left(x\right) is decreasing on the interval
Critical Points and Extrema
The following is the graph of f′′(x)f''\left(x\right), which of the following statements (if any) is correct?
Intervals of Increase and Decrease: Rate of Change
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Intervals of Increase and Decrease
f′(x)f'(x) plot for a differentiable function is depicted as below. Find interval(s) for which f(x)f\left(x\right) is increasing.
Concavity and Inflection Points
Find the intervals of increase/decrease and the intervals of concavity for f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Concavity, Inflection Points, Extrema and Intervals
Given f(x)=2−2x+13x3f(x)=2-2x+\frac{1}{3}x^3, find
the intervals of increase and decrease
the local maximum and minimum values
Increasing Interval: Relative extrema
If the derivative of the function ff is f′(x)=ex(x−3)2(x+1)(2−x)f'\left(x\right)=e^x\left(x-3\right)^2\left(x+1\right)\left(2-x\right), then f(x)f\left(x\right) is increasing on the interval?
Intervals of Increase and Decrease
Find the intervals in which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Q.\textbf{Q.} Find the absolute extrema of f(x)=2xx2+1\displaystyle f(x)=\frac{2x}{x^2+1} on the interval [0,2][0,2]. Then find the local/absolute extrema on its domain, and intervals of increase and decrease.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=x3+3x2−9x+1f(x)=x^{3}+3x^{2}-9x+1 is increasing or decreasing.
Intervals of Increase and Decrease
Find the intervals on which the function f(x)=−1x+1\displaystyle f(x)=-\frac{1}{x+1} is increasing or decreasing.
Q:\bf{Q:} Which of the following functions is/are always increasing?
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x2−2ln⁡xf(x)=x^2-2\ln{x}
f′(x)f'(x) plot for a differentiable function is depicted below.
Find interval(s) at which ff is increasing.
f′(x)f'(x) plot for a differentiable function is depicted below.
a) Find all critical point of ff in the range shown in the plot.
b) Determine which of the critical points found in the previous part are local max/min.
Intervals of increase and decrease
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
f(x)=x3ln⁡xf(x) = x^3 \ln x
Optimization
For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.
y=e4x2xy=\frac{e^{4x}}{2x}
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Intervals of increase and decrease
If f′(x)=x5x+1(x−1)3(x+2)f'\left(x\right)=x^5\sqrt{x+1}\left(x-1\right)^3\left(x+2\right), which of the following statements about the graph of f(x)f\left(x\right) must be true?
(Select all that apply)
Exercises
Find the intervals in which the function f(x)=−1x+1\displaystyle f(x)=\frac{-1}{x+1} is increasing or decreasing.
f′(x)f'(x) plot for a differentiable function is depicted as below. Answer the following questions.
Find interval(s) at which ffis increasing.
More Logarithmic Functions Questions:
Find the inverse of the function y=log⁡5(2x−1)−9y=\log_5\left(2x-1\right)-9.
Find the inverse of the function y=log⁡5(2x−1)−9y=\log_5\left(2x-1\right)-9.
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Logarithmic Functions
Find the domain of the following function:
f(x)=x2−2x+1ln⁡(x2−5x+6)f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Practice: Solving a logarithmic function
Solve for 𝑥 in the equation 2ln⁡(x−1)−ln⁡(4)=02\ln\left(x-1\right)-\ln\left(4\right)=0
Inverse of Logarithmic Functions
Find the inverse of y=5ln⁡(3ln⁡x−2)y=5\ln\left(3\ln x-2\right)
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Logarithmic Functions
Solve for xx in the equation: eln⁡(1+e−x)=e3e^{\ln\left(1+e^{-x}\right)}=e^3
Logarithmic Functions
Solve for xx in the equation: eln⁡(1+e−x)=e3e^{\ln\left(1+e^{-x}\right)}=e^3
Solving exponential functions
Solve for x if 32x/3=8x−1232^{x/3} = 8^{x-12}
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Logarithmic Functions
Find the inverse function of f(x)=2ln⁡(x+e) f\left(x\right)=2\ln\left(x+e\right)\ .
Logarithmic Functions
Find the inverse function of f(x)=2ln⁡(x+e) f\left(x\right)=2\ln\left(x+e\right)\ .
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Evaluate 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Evaluate 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Evaluate 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Domain of Log Functions
Determine the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Logarithmic Functions
Solve for xx in the equation: eln⁡(1+e−x)=e3e^{\ln\left(1+e^{-x}\right)}=e^3
Solving exponential functions
Solve for x if 32x/3=8x−1232^{x/3} = 8^{x-12}
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
Additional Practice Problems
Simplify the following expressions:
a. ln⁡(3e2)+e2ln⁡3\ln\left(3e^2\right)+e^{2\ln3}
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
Logarithmic Functions: Log Values
Practice: Log Values
If x=log⁡218x=\log_2\sqrt{\frac{1}{8}} , y=ln⁡(10e2)+ln⁡(0.1)y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right), and z=e−2ln⁡3z=e^{-2\ln3}, which of the following statements is true?
Q:\bf{Q:} Find the inverse function of 𝑓(𝑥)=2ln⁡(𝑥+e) 𝑓(𝑥) = 2 \ln (𝑥 + e)
Q:\bf{Q:} Find the inverse of f(x)=ln⁡(x−4)f\left(x\right)=\ln\left(x-4\right) and state the domain and range for both ff and f−1f^{-1} .
Practice: Combining Logs
Q:\bf{Q:} Evaluate log⁡2 116−log⁡215+log⁡230\log_2\ \dfrac{1}{16}-\log_215+\log_230
Logarithmic Functions
Find the domain of the following function: f(x)=1−xlog⁡(1+x)f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
Logarithmic Functions: Determine the domain
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Logarithmic Functions
Find the domain of the function f(x)=ln⁡[ln⁡(x2)]f\left(x\right)=\ln\left[\ln\left(x^2\right)\right].
Logarithmic Functions
Solve for xx in the equation 2ln⁡(x−1)−ln⁡(4)=02\ln\left(x-1\right)-\ln\left(4\right)=0
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Inverse of Logarithmic Functions
Find the inverse of y=5ln⁡(3ln⁡x−2)y=5\ln\left(3\ln x-2\right)
Range of Inverse
Find the range of the inverse of y=x+2+ln⁡(5−x)y=\sqrt{x+2}+\ln(5-x) .
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither: f(x)=log⁡31−x1+xf\left(x\right)=\log^3\dfrac{1-x}{1+x}
Q:\bf{Q:} State the possible values of ff and gg given f(g(x))=x−ln⁡xf\left(g\left(x\right)\right)=\sqrt{x-\ln x}
Logarithmic Functions
Find the domain of f(x)=[ln⁡(x−5)−2]34f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
Inequalities with Logs
Find all xx that satisfy the inequality ln⁡(2x−1)>e3\ln\left(2x-1\right)>e^3
Logarithmic and Trigonometric Functions
Solve for x in the equation ln⁡(4−4sin⁡2x)=0\ln\left(4-4\sin^2x\right)=0
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Logarithmic Functions
Solve for xx in the equation: eln⁡(1+e−x)=e3e^{\ln\left(1+e^{-x}\right)}=e^3
Rules of Logs
Expand the expression log⁡[x3sin⁡(x4)cos⁡2(5x)(102x)]\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate each logarithm and match it to its corresponding answer.
Logarithmic Functions
If log⁡3=a\log{3}=a and log⁡5=b\log{5}=b, then determine the following in terms of a and b:
Solving Logarithmic Equations
Solve log⁡2(2x+1)−log⁡2(x−1)=3\log_2{(2x+1)}-\log_2{(x-1)}=3.
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If log⁡7343=b−a\log_{7}{343}=b-a and log⁡636=2a+2b\log_{6}{36}=2a+2b, what is aa and bb?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If log⁡5625=a+b\log_{5}{625}=a+b and log⁡327=2a−b\log_{3}{27}=2a-b, what is aa and bb?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log⁡56+log⁡52x2=log⁡548 \log_5{6} + \log_5{2x^2}=\log_5{48}
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log⁡(x−3)−log⁡(x−5)=log⁡5\log(x-3)-\log(x-5)=\log5
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log⁡82+log⁡84x2=1\log_82+\log_84x^2=1
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log⁡9(x+6)−log⁡9x=log⁡92\log_9(x+6)-\log_9x=\log_92
Logarithm Laws
Practice: Logarithm Laws
If log⁡50=9.5\log50=9.5, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If log⁡20=3.4872\log20=3.4872, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If log⁡5=a\log5=a and log⁡7=b\log7=b, write each logarithm in terms of aa and bb.
Logarithm Laws
Practice: Logarithm Laws
If log⁡3=a\log3=a and log⁡5=b\log5=b, write each logarithm in terms of aa and bb.
Logarithm Laws
Practice: Logarithm Laws
Evaluate log⁡3(812734353965)\log_{3}{\Bigg(\dfrac{81\sqrt[4]{27^3}3^{\frac{5}{3}}}{9^{\frac{6}{5}}}\Bigg)}. Leave answer in exact form.
Logarithm Laws
Practice: Logarithm Laws
Which of the following is equivalent to log⁡a(a4b3c5d32e2)\log_{a}{\Bigg(\dfrac{a^4\sqrt[3]{b}c^5}{d^{\frac{3}{2}}e^2}\Bigg)}?
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate log⁡1162=128\log_{\frac{1}{16}}{^2=\sqrt{128}}.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate log⁡19327\log_{\frac{1}{9}}{^3\sqrt{27}}.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate each logarithm and match it to its corresponding answer.
Basics of Logarithms
Practice: Basics of Logarithms
Match each exponential expression to its inverse.
Basics of Logarithms
Practice: Basics of Logarithms
Match each exponential expression to its inverse.
Question 1
Suppose that you have conducted an experiment resulting in the data plotted below.
The following functions f(x)f(x) depend on a parameter b>0b > 0. Which function would you use to fit the data points?
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Transforming a Log
Given f(x)=log⁡3(2x+4)f\left(x\right)=\log_3\left(2x+4\right) , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for x:
a. 9x−2=(127)2x+19^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
Find the domain of the function f(x)=8−4ln⁡(x)f\left(x\right)=\sqrt{8-4\ln\left(x\right)}
When we plot the function y=10000x3y=10000x^3 on a log-log graph, we get
Find the domain of the function f(x)=ln⁡([[x]]−4)f\left(x\right)=\ln\left(\left[\left[x\right]\right]-4\right) where [[x]]\left[\left[x\right]\right] represents the greatest integer of x.
Logarithmic Functions
Find the inverse of the function f(x)=2−e3x+1f(x) = 2-e^{3x+1} and find its domain.
Logarithmic Functions
Find the inverse of f(x)=2ln⁡(x+e)f(x) = 2\ln(x+e)
Logarithmic Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f(x) = \ln(\ln(x^2))
Solving exponential functions
Solve for x if 32x/3=8x−1232^{x/3} = 8^{x-12}
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for x:
a. 9x−2=(127)2x+19^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation 2ln⁡(x−1)−ln⁡(4)=02\ln\left(x-1\right)-\ln\left(4\right)=0
Enter your answers as numbers only. If there is more than 1 answer for x, type them from smallest to biggest, separated by a comma with no spaces in between (for example, -5,1,2)
Logarithmic Functions
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
12[log⁡3(x−1)+log⁡3(x−3)]\frac{1}{2}\left[\log_3\left(x-1\right)+\log_3\left(x-3\right)\right]
Find the inverse of the function y=log⁡5(2x−1)−9y=\log_5\left(2x-1\right)-9.
Logarithmic Functions
Solve for xx in the equation: eln⁡(1+e−x)=e3e^{\ln\left(1+e^{-x}\right)}=e^3
Rules of Logs
Expand the expression log⁡[x3sin⁡(x4)cos⁡2(5x)(102x)]\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
Logarithmic and Trigonometric Functions
Solve for x in the equation ln⁡(4−4sin⁡2x)=0\ln\left(4-4\sin^2x\right)=0
Inequalities with Logs
Find all xx that satisfy the inequality ln⁡(2x−1)>e3\ln\left(2x-1\right)>e^3
Inverse of Logarithmic Functions
Find the inverse of y=5ln⁡(3ln⁡x−2)y=5\ln\left(3\ln x-2\right)
Inverse of Exponent
Find the inverse of the given function, and its domain and range: f(x)=5e2x3−1−4f\left(x\right)=5e^{2x^3-1}-4
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither: f(x)=log⁡31−x1+xf\left(x\right)=\log^3\dfrac{1-x}{1+x}
Logarithmic Functions
Find the domain of f(x)=[ln⁡(x−5)−2]34f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
Logarithmic Functions: Determine the domain
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Logarithmic Functions
Find the domain of the function f(x)=ln⁡[ln⁡(x2)]f\left(x\right)=\ln\left[\ln\left(x^2\right)\right].
Logarithmic Functions
Solve for xx in the equation 2ln⁡(x−1)−ln⁡(4)=02\ln\left(x-1\right)-\ln\left(4\right)=0
Logarithmic Functions
Find the domain of the following function: f(x)=1−xlog⁡(1+x)f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
Q:\bf{Q:} Find the inverse function of 𝑓(𝑥)=2ln⁡(𝑥+e) 𝑓(𝑥) = 2 \ln (𝑥 + e)
Range of Inverse
Find the range of the inverse of y=x+2+ln⁡(5−x)y=\sqrt{x+2}+\ln(5-x) .
Q:\bf{Q:} State the possible values of ff and gg given f(g(x))=x−ln⁡xf\left(g\left(x\right)\right)=\sqrt{x-\ln x}
Q:\bf{Q:} Find the inverse of f(x)=ln⁡(x−4)f\left(x\right)=\ln\left(x-4\right) and state the domain and range for both ff and f−1f^{-1} .
Practice: Combining Logs
Q:\bf{Q:} Evaluate log⁡2 116−log⁡215+log⁡230\log_2\ \dfrac{1}{16}-\log_215+\log_230
What value of xx solves the equation
log⁡2(x)+log⁡2(x−2)=3\log_2\left(x\right)+\log_2\left(x-2\right)=3 ?
Logarithmic Functions
Consider the functions u(t)=(t+2)2u(t)=(t+2)^2 and v(t)=ln⁡(t)v(t)=\ln(\sqrt t). Find (v∘u)(t)(v \circ u)(t) and the domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
3−log⁡3(1x2)\displaystyle3^{-\log_3\left(\frac{1}{x^2}\right)}
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
eln⁡(ln⁡ e7)+ln⁡(eeln⁡6)e^{\ln\left(\ln\ e^7\right)}+\ln\left(e^{e^{\ln6}}\right)
Logarithmic Functions
Simplify log⁡78log⁡732\frac{\log_78}{\log_732}
Logarithmic Functions
Evaluate e[2ln⁡4−ln⁡2]+ln⁡(e3e2)e^{[2\ln4-\ln2]}+\ln(e^3e^2).
Logarithmic Functions
Find the inverse of f(x)=ln⁡(x−1x+1)f\left(x\right)=\ln\left(\frac{x-1}{x+1}\right)
Logarithmic Functions
Find the domain of the following function:
f(x)=x2−2x+1ln⁡(x2−5x+6)f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
Practice: Inverse function
Find the inverse of f(x)=ln⁡18−2−3cos⁡xf\left(x\right)=\ln\frac{1}{\sqrt{8-2^{-3\cos x}}}, 0≤x<π0 \leq x < \pi.
Logarithmic Functions
Solve for xx in the equation ln⁡(ln⁡(x+1))=0\ln\left(\ln\left(x+1\right)\right)=0
Logarithmic Functions
Solve for xx in the equation ln⁡(3x−1)=−e\ln\left(3x-1\right)=-e.
Logarithmic Functions
Find the inverse function of f(x)=2ln⁡(x+e) f\left(x\right)=2\ln\left(x+e\right)\ .
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Solve for xx in the equation ln⁡(ln⁡(x+1))=0\ln\left(\ln\left(x+1\right)\right)=0
Practice: Log & Exponential Function Properties
Practice: Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Practice: Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
a. f−1(x)=ln⁡(3x−1)−2f^{-1}\left(x\right)=\ln\left(3x-1\right)-2; domain of f−1f^{-1} is (13,∞)\left(\frac{1}{3},\infty\right)
Practice: Finding an inverse function
Practice Question: Finding an Inverse Function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Practice: Domain of Log Functions
Practice: Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation 2ln⁡(x−1)−ln⁡(4)=02\ln\left(x-1\right)-\ln\left(4\right)=0
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Logarithmic Functions
Find the domain of the function f(x)=ln⁡(x2−5)f\left(x\right)=\ln\left(x^2-5\right).
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Domain of Log Functions
Find the domain of the function f(x)=ln⁡(ln⁡(x2))f\left(x\right)=\ln\left(\ln\left(x^2\right)\right).
Logarithmic Functions: Domain and Range
Find the formula of the function obtained by shifting the graph of y=ln⁡xy=\ln x 2 units down, then reflecting the graph about the x-axis, and finally, shifting the graph 1 unit up. State the domain and range of the function.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
b. Simplify 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Logarithmic Functions: Log Values
Practice: Log Values
If x=log⁡218x=\log_2\sqrt{\frac{1}{8}} , y=ln⁡(10e2)+ln⁡(0.1)y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right), and z=e−2ln⁡3z=e^{-2\ln3}, which of the following statements is true?
Solving Log Equations
If e⋅ln⁡(1+ex)=1e\cdot\ln\left(1+e^x\right)=1, solve for xx
More Exponential Functions Questions:
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Exponential Functions
If 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}, find xx
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Properties of Exponential Functions
If f(x)=−4⋅e3xπ⋅22xf\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}, which of the following statements is true?
Exponential Functions: Domain & Range
Find the domain and range of f(x)=2−9x−8f\left(x\right)=\sqrt{2^{-9x}-8}.
Exponential Functions
Which of the following statements is/are true about the exponential function f(x)=C4x+bf\left(x\right)=C4^x+b whose graph passes through the points (1,1)\left(1,1\right)and (12,2)\left(\frac{1}{2},2\right)?
i. f(x)=2(4x)−1f\left(x\right)=2\left(4^x\right)-1
ii. It has a horizontal asymptote at y=−1y=-1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis
Practice: Exponents Equation
Q:\bf{Q:} Solve for xx in 2x2−x=642^{x^2-x}=64
Exponential Functions
Find the domain of the function f(x)=1(ex−1)(x+1)+12−x2f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Exponential Functions: Inequalities with Exponents
Find all xx that satisfy the inequality ex>e−xe^x>e^{−x}
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Solving Exponential Equations
Solve 5(1128)2x+5=10(32)x+15\Bigg(\dfrac{1}{128}\Bigg)^{2x+5}=10(32)^{x+1}
Simplifying Exponential Expressions
Simplify (72ab(2a+b)32a−3b)\Bigg(\dfrac{72^{ab}(2^{a+b})}{3^{2a-3b}}\Bigg)
Solving Exponential Equations
Solve 4(8)3x+1=32(4)x+24(8)^{3x+1}=32(4)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 4x−6(2x)+8=04^x-6(2^x)+8=0
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 32x−3x=03^{2x}-3^x=0
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(216125)3x=(3625)2x−1\Bigg(\dfrac{216}{125}\Bigg)^{3x}=\Bigg(\dfrac{36}{25}\Bigg)^{2x-1}
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(34)2x−1=(81256)x+2\Bigg(\dfrac{3}{4}\Bigg)^{2x-1}=\Bigg(\dfrac{81}{256}\Bigg)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 254x+5=125x−325^{4x+5}=125^{x-3}. Leave answer as a fraction.
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 42x+1=83x−14^{2x+1}=8^{3x-1}. Leave answer as a fraction.
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions: Domain & Range
Find the domain and range of f(x)=2−9x−8f\left(x\right)=\sqrt{2^{-9x}-8}.
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Exponential Functions
Find the domain of the function f(x)=e2x−8f\left(x\right)=e^{\sqrt{2x-8}}
Transformations
Which of the following is the equation of the function produced by vertically shifting the graph of y=exy=e^x up 3 units and then vertically stretching it by a factor of 4?
Q:\bf{Q:} The lifespan of a mammal species is related to its heart rate by the function L(h)=120h−0.01L\left(h\right)=120h^{-0.01} where LL is the lifespan in years and hh is the heart rate in beats per minute. Which of the following observations are true? Select all that apply.
Note: This question is for practice purposes only, it is not based on real scientific data.
Which of this functions represents a quantity that doubles every 33 years?
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Exponential Functions: Inequalities with Exponents
Find all xx that satisfy the inequality ex>e−xe^x>e^{−x}
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Exponential Functions
Find the domain of the function f(x)=1(ex−1)(x+1)+12−x2f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
Practice: Exponents Equation
Q:\bf{Q:} Solve for xx in 2x2−x=642^{x^2-x}=64
Practice: Exponential function
The exponential function y=bx+k, (b>0)y=b^x+k,\ \left(b>0\right) passes through the point (2,1316)\left(2,\frac{13}{16}\right) and has a horizontal asymptote at y=14y=\frac{1}{4}.
Find the equation of this graph.
Exponential Functions
Solve the equation for xx .
e2x=11ex−30e^{2x}=11e^x-30
Exponential Functions
If h(x)=ex+2x+1h\left(x\right)=e^x+2x+1, find the value of h−1(2)h^{-1}\left(2\right)
Exponential Functions
Find the domain of the function f(x)=18−2−3xf\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
Practice: Log & Exponential Function Properties
Practice: Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions
The exponential function y=bx+k, (b>0)y=b^x+k,\ \left(b>0\right) passes through the point (2,1316)\left(2,\frac{13}{16}\right) and has a horizontal asymptote at y=14y=\frac{1}{4}.
Find the equation of this graph.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Exponential Functions
Find the domain of the function f(x)=18−2−3xf\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
Properties of Exponential Functions
If f(x)=−4⋅e3xπ⋅22xf\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}, which of the following statements is true?
Exponential Functions
Which of the following statements is/are true about the exponential function f(x)=C4x+bf\left(x\right)=C4^x+b whose graph passes through the points (1,1)\left(1,1\right)and (12,2)\left(\frac{1}{2},2\right)?
i. f(x)=2(4x)−1f\left(x\right)=2\left(4^x\right)-1
ii. It has a horizontal asymptote at y=−1y=-1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis