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

Logarithmic Functions

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Logarithmic Functions

For an exponential ax,(a>0)a^x,(a>0) to have an inverse function, we needa≠1a ≠ 1. In this case, the inverse function is f(x)=log⁡ax,f(x)=\log _ax,which has domain x>0x>0, is defined by the following relationship
log⁡ax=y  ⟺  ay=x\boxed{\log_{a} x=y \iff a^y=x}


f(x)f(x) has a vertical asymptote at
x=0

The Natural Logarithm has a base of ee: ln⁡x=log⁡ex\ln x =\log_e x

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Properties of Logarithms

For x,y,c∈I ⁣R  and  x,y>0x, y, c\in\R \;\text{and} \;x,y>0
  1. log⁡1=0\log1=0
  2. log⁡(xy)=log⁡(x)+log⁡(y)\log(xy) = \log(x) + \log(y)
  3. log⁡(xy)=log⁡(x)−log⁡(y)\log\left( \frac{x}{y} \right) = \log(x) - \log(y)
  4. log⁡(xc)=clog⁡(x)\log(x^c) = c\log(x)
  5. log⁡aax=x\log_aa^x = x
  6. alog⁡ax=xa^{\log_ax} = x
  7. log⁡cx=ln⁡xln⁡c\displaystyle \log_c x=\frac{\ln x}{\ln c}
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Example: Evaluating Logarithms

Evaluate log⁡2 116−log⁡215+log⁡230\log_2\ \frac{1}{16}-\log_215+\log_230

Let x=log⁡2116−log⁡215+log⁡230x=\log_2\frac{1}{16}-\log_215+\log_230
Using log rules, we can combine the right hand side
x=log⁡2(116÷15×30)x=\log_2\left(\frac{1}{16}\div15\times30\right)
x=log⁡2(116×15×30)x=\log_2\left(\frac{1}{16\times15}\times30\right)
x=log⁡2(18)x=\log_2\left(\frac{1}{8}\right)

Using the relationship between log and exponent, we can rewrite this
2x=182^x=\frac{1}{8}
2x=2−32^x=2^{-3}
So, x=−3x=-3.
Therefore, the expression evaluates to −3-3
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Example: Inverse of Logarithms

Find the inverse of f(x)=ln⁡(x−4)f\left(x\right)=\ln\left(x-4\right) and state the domain and range of both f(x)f(x) and f−1(x)f^{-1}(x).

Steps:
1. Write y=f(x)y=f\left(x\right)
y=ln⁡(x−4)y=\ln\left(x-4\right)

2. Solve for xx
ey=eln⁡(x−4)e^y=e^{\ln\left(x-4\right)}
ey=x−4e^y=x-4
ey+4=xe^y+4=x
x=ey+4x=e^y+4

3. Swap xx and yy
y=ex+4y=e^x+4

4. Replace yy with f−1(x)f^{-1}\left(x\right), this is the inverse function
Therefore, f−1(x)=ex+4f^{-1}\left(x\right)=e^x+4

Domain and range:
𝑓 has domain (4,∞)\left(4,\infty\right) and range (−∞,∞)\left(-\infty,\infty\right)
f−1f^{-1} has domain (−∞,∞)\left(-\infty,\infty\right) and range (4,∞)\left(4,\infty\right)
Solve for xx in the equation 2ln⁡x=ln⁡4+ln⁡(x−1)2\ln x=\ln4+\ln\left(x-1\right).

Extra Practice
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).
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
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