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Logarithmic Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Logarithmic Functions
4 Activities
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
−
ln
(
e
3
−
1
)
-\ln\left(e^3-1\right)
−
ln
(
e
3
−
1
)
ln
(
e
3
−
1
)
\ln\left(e^3-1\right)
ln
(
e
3
−
1
)
ln
(
e
3
)
\ln\left(e^3\right)
ln
(
e
3
)
ln
(
e
3
)
−
ln
1
\ln\left(e^3\right)-\ln1
ln
(
e
3
)
−
ln
1
I don't know
Check Submission
More Logarithmic Functions Questions:
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
x
2
−
2
x
+
1
ln
(
x
2
−
5
x
+
6
)
f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
f
(
x
)
=
ln
(
x
2
−
5
x
+
6
)
x
2
−
2
x
+
1
Transforming a Log
Given
f
(
x
)
=
log
3
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, 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
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Practice: Solving a logarithmic function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Domain of Log Functions
Determine the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
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
(
3
e
2
)
+
e
2
ln
3
\ln\left(3e^2\right)+e^{2\ln3}
ln
(
3
e
2
)
+
e
2
l
n
3
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x
Logarithmic Functions: Log Values
Practice: Log Values
If
x
=
log
2
1
8
x=\log_2\sqrt{\frac{1}{8}}
x
=
lo
g
2
8
1
,
y
=
ln
(
10
e
2
)
+
ln
(
0.1
)
y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right)
y
=
ln
(
e
2
10
)
+
ln
(
0.1
)
, and
z
=
e
−
2
ln
3
z=e^{-2\ln3}
z
=
e
−
2
l
n
3
, which of the following statements is true?
Q
:
\bf{Q:}
Q
:
Find the inverse function of
𝑓
(
𝑥
)
=
2
ln
(
𝑥
+
e
)
𝑓(𝑥) = 2 \ln (𝑥 + e)
f
(
x
)
=
2
ln
(
x
+
e
)
Q
:
\bf{Q:}
Q
:
Find the inverse of
f
(
x
)
=
ln
(
x
−
4
)
f\left(x\right)=\ln\left(x-4\right)
f
(
x
)
=
ln
(
x
−
4
)
and state the domain and range for both
f
f
f
and
f
−
1
f^{-1}
f
−
1
.
Practice: Combining Logs
Q
:
\bf{Q:}
Q
:
Evaluate
log
2
1
16
−
log
2
15
+
log
2
30
\log_2\ \dfrac{1}{16}-\log_215+\log_230
lo
g
2
16
1
−
lo
g
2
15
+
lo
g
2
30
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
1
−
x
log
(
1
+
x
)
f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
f
(
x
)
=
lo
g
(
1
+
x
)
1
−
x
Logarithmic Functions: Determine the domain
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
[
ln
(
x
2
)
]
f\left(x\right)=\ln\left[\ln\left(x^2\right)\right]
f
(
x
)
=
ln
[
ln
(
x
2
)
]
.
Logarithmic Functions
Solve for
x
x
x
in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Range of Inverse
Find the range of the inverse of
y
=
x
+
2
+
ln
(
5
−
x
)
y=\sqrt{x+2}+\ln(5-x)
y
=
x
+
2
+
ln
(
5
−
x
)
.
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither:
f
(
x
)
=
log
3
1
−
x
1
+
x
f\left(x\right)=\log^3\dfrac{1-x}{1+x}
f
(
x
)
=
lo
g
3
1
+
x
1
−
x
Q
:
\bf{Q:}
Q
:
State the possible values of
f
f
f
and
g
g
g
given
f
(
g
(
x
)
)
=
x
−
ln
x
f\left(g\left(x\right)\right)=\sqrt{x-\ln x}
f
(
g
(
x
)
)
=
x
−
ln
x
Logarithmic Functions
Find the domain of
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
3
4
f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
4
3
Inequalities with Logs
Find all
x
x
x
that satisfy the inequality
ln
(
2
x
−
1
)
>
e
3
\ln\left(2x-1\right)>e^3
ln
(
2
x
−
1
)
>
e
3
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Transforming a Log
Given
f
(
x
)
=
log
3
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Rules of Logs
Expand the expression
log
[
x
3
sin
(
x
4
)
cos
2
(
5
x
)
(
10
2
x
)
]
\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
lo
g
[
cos
2
(
5
x
)
(
1
0
2
x
)
x
3
sin
(
x
4
)
]
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
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
lo
g
3
=
a
and
log
5
=
b
\log{5}=b
lo
g
5
=
b
, then determine the following in terms of a and b:
Solving Logarithmic Equations
Solve
log
2
(
2
x
+
1
)
−
log
2
(
x
−
1
)
=
3
\log_2{(2x+1)}-\log_2{(x-1)}=3
lo
g
2
(
2
x
+
1
)
−
lo
g
2
(
x
−
1
)
=
3
.
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If
log
7
343
=
b
−
a
\log_{7}{343}=b-a
lo
g
7
343
=
b
−
a
and
log
6
36
=
2
a
+
2
b
\log_{6}{36}=2a+2b
lo
g
6
36
=
2
a
+
2
b
, what is
a
a
a
and
b
b
b
?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
If
log
5
625
=
a
+
b
\log_{5}{625}=a+b
lo
g
5
625
=
a
+
b
and
log
3
27
=
2
a
−
b
\log_{3}{27}=2a-b
lo
g
3
27
=
2
a
−
b
, what is
a
a
a
and
b
b
b
?
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve.
log
5
6
+
log
5
2
x
2
=
log
5
48
\log_5{6} + \log_5{2x^2}=\log_5{48}
lo
g
5
6
+
lo
g
5
2
x
2
=
lo
g
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
lo
g
(
x
−
3
)
−
lo
g
(
x
−
5
)
=
lo
g
5
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log
8
2
+
log
8
4
x
2
=
1
\log_82+\log_84x^2=1
lo
g
8
2
+
lo
g
8
4
x
2
=
1
Solving Logarithmic Equations
Practice: Solving Logarithmic Equations
Solve and check.
log
9
(
x
+
6
)
−
log
9
x
=
log
9
2
\log_9(x+6)-\log_9x=\log_92
lo
g
9
(
x
+
6
)
−
lo
g
9
x
=
lo
g
9
2
Logarithm Laws
Practice: Logarithm Laws
If
log
50
=
9.5
\log50=9.5
lo
g
50
=
9.5
, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If
log
20
=
3.4872
\log20=3.4872
lo
g
20
=
3.4872
, find an approximation for the following:
Logarithm Laws
Practice: Logarithm Laws
If
log
5
=
a
\log5=a
lo
g
5
=
a
and
log
7
=
b
\log7=b
lo
g
7
=
b
, write each logarithm in terms of
a
a
a
and
b
b
b
.
Logarithm Laws
Practice: Logarithm Laws
If
log
3
=
a
\log3=a
lo
g
3
=
a
and
log
5
=
b
\log5=b
lo
g
5
=
b
, write each logarithm in terms of
a
a
a
and
b
b
b
.
Logarithm Laws
Practice: Logarithm Laws
Evaluate
log
3
(
81
27
3
4
3
5
3
9
6
5
)
\log_{3}{\Bigg(\dfrac{81\sqrt[4]{27^3}3^{\frac{5}{3}}}{9^{\frac{6}{5}}}\Bigg)}
lo
g
3
(
9
5
6
81
4
2
7
3
3
3
5
)
. Leave answer in exact form.
Logarithm Laws
Practice: Logarithm Laws
Which of the following is equivalent to
log
a
(
a
4
b
3
c
5
d
3
2
e
2
)
\log_{a}{\Bigg(\dfrac{a^4\sqrt[3]{b}c^5}{d^{\frac{3}{2}}e^2}\Bigg)}
lo
g
a
(
d
2
3
e
2
a
4
3
b
c
5
)
?
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate
log
1
16
2
=
128
\log_{\frac{1}{16}}{^2=\sqrt{128}}
lo
g
16
1
2
=
128
.
Basics of Logarithms
Practice: Basics of Logarithms
Evaluate
log
1
9
3
27
\log_{\frac{1}{9}}{^3\sqrt{27}}
lo
g
9
1
3
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)
f
(
x
)
depend on a parameter
b
>
0
b > 0
b
>
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
(
2
x
+
4
)
f\left(x\right)=\log_3\left(2x+4\right)
f
(
x
)
=
lo
g
3
(
2
x
+
4
)
, 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.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Find the domain of the function
f
(
x
)
=
8
−
4
ln
(
x
)
f\left(x\right)=\sqrt{8-4\ln\left(x\right)}
f
(
x
)
=
8
−
4
ln
(
x
)
When we plot the function
y
=
10000
x
3
y=10000x^3
y
=
10000
x
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)
f
(
x
)
=
ln
(
[
[
x
]
]
−
4
)
where
[
[
x
]
]
\left[\left[x\right]\right]
[
[
x
]
]
represents the greatest integer of x.
Logarithmic Functions
Find the inverse of the function
f
(
x
)
=
2
−
e
3
x
+
1
f(x) = 2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
and find its domain.
Logarithmic Functions
Find the inverse of
f
(
x
)
=
2
ln
(
x
+
e
)
f(x) = 2\ln(x+e)
f
(
x
)
=
2
ln
(
x
+
e
)
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f(x) = \ln(\ln(x^2))
f
(
x
)
=
ln
(
ln
(
x
2
))
Solving exponential functions
Solve for
x
if
32
x
/
3
=
8
x
−
12
32^{x/3} = 8^{x-12}
3
2
x
/3
=
8
x
−
12
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
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
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
1
2
[
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]
2
1
[
lo
g
3
(
x
−
1
)
+
lo
g
3
(
x
−
3
)
]
Find the inverse of the function
y
=
log
5
(
2
x
−
1
)
−
9
y=\log_5\left(2x-1\right)-9
y
=
lo
g
5
(
2
x
−
1
)
−
9
.
Logarithmic Functions
Solve for
x
x
x
in the equation:
e
ln
(
1
+
e
−
x
)
=
e
3
e^{\ln\left(1+e^{-x}\right)}=e^3
e
l
n
(
1
+
e
−
x
)
=
e
3
Rules of Logs
Expand the expression
log
[
x
3
sin
(
x
4
)
cos
2
(
5
x
)
(
10
2
x
)
]
\log\left[\dfrac{x^3\sin\left(x^4\right)}{\cos^2\left(5x\right)\left(10^{2x}\right)}\right]
lo
g
[
cos
2
(
5
x
)
(
1
0
2
x
)
x
3
sin
(
x
4
)
]
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Inequalities with Logs
Find all
x
x
x
that satisfy the inequality
ln
(
2
x
−
1
)
>
e
3
\ln\left(2x-1\right)>e^3
ln
(
2
x
−
1
)
>
e
3
Inverse of Logarithmic Functions
Find the inverse of
y
=
5
ln
(
3
ln
x
−
2
)
y=5\ln\left(3\ln x-2\right)
y
=
5
ln
(
3
ln
x
−
2
)
Inverse of Exponent
Find the inverse of the given function, and its domain and range:
f
(
x
)
=
5
e
2
x
3
−
1
−
4
f\left(x\right)=5e^{2x^3-1}-4
f
(
x
)
=
5
e
2
x
3
−
1
−
4
Even or Odd Logarithmic Functions
Determine if the following function is even, odd, or neither:
f
(
x
)
=
log
3
1
−
x
1
+
x
f\left(x\right)=\log^3\dfrac{1-x}{1+x}
f
(
x
)
=
lo
g
3
1
+
x
1
−
x
Logarithmic Functions
Find the domain of
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
3
4
f\left(x\right)=\left[\ln\left(x-5\right)-2\right]^{^{\frac{3}{4}}}
f
(
x
)
=
[
ln
(
x
−
5
)
−
2
]
4
3
Logarithmic Functions: Determine the domain
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
[
ln
(
x
2
)
]
f\left(x\right)=\ln\left[\ln\left(x^2\right)\right]
f
(
x
)
=
ln
[
ln
(
x
2
)
]
.
Logarithmic Functions
Solve for
x
x
x
in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
1
−
x
log
(
1
+
x
)
f(x)=\dfrac{\sqrt{1-x}}{\log\left(1+x\right)}
f
(
x
)
=
lo
g
(
1
+
x
)
1
−
x
Q
:
\bf{Q:}
Q
:
Find the inverse function of
𝑓
(
𝑥
)
=
2
ln
(
𝑥
+
e
)
𝑓(𝑥) = 2 \ln (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)
y
=
x
+
2
+
ln
(
5
−
x
)
.
Q
:
\bf{Q:}
Q
:
State the possible values of
f
f
f
and
g
g
g
given
f
(
g
(
x
)
)
=
x
−
ln
x
f\left(g\left(x\right)\right)=\sqrt{x-\ln x}
f
(
g
(
x
)
)
=
x
−
ln
x
Q
:
\bf{Q:}
Q
:
Find the inverse of
f
(
x
)
=
ln
(
x
−
4
)
f\left(x\right)=\ln\left(x-4\right)
f
(
x
)
=
ln
(
x
−
4
)
and state the domain and range for both
f
f
f
and
f
−
1
f^{-1}
f
−
1
.
Practice: Combining Logs
Q
:
\bf{Q:}
Q
:
Evaluate
log
2
1
16
−
log
2
15
+
log
2
30
\log_2\ \dfrac{1}{16}-\log_215+\log_230
lo
g
2
16
1
−
lo
g
2
15
+
lo
g
2
30
What value of
x
x
x
solves the equation
log
2
(
x
)
+
log
2
(
x
−
2
)
=
3
\log_2\left(x\right)+\log_2\left(x-2\right)=3
lo
g
2
(
x
)
+
lo
g
2
(
x
−
2
)
=
3
?
Logarithmic Functions
Consider the functions
u
(
t
)
=
(
t
+
2
)
2
u(t)=(t+2)^2
u
(
t
)
=
(
t
+
2
)
2
and
v
(
t
)
=
ln
(
t
)
v(t)=\ln(\sqrt t)
v
(
t
)
=
ln
(
t
)
. Find
(
v
∘
u
)
(
t
)
(v \circ u)(t)
(
v
∘
u
)
(
t
)
and the domain.
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
3
−
log
3
(
1
x
2
)
\displaystyle3^{-\log_3\left(\frac{1}{x^2}\right)}
3
−
l
o
g
3
(
x
2
1
)
Logarithmic Functions
Simplify the following expression using the laws of logarithm.
e
ln
(
ln
e
7
)
+
ln
(
e
e
ln
6
)
e^{\ln\left(\ln\ e^7\right)}+\ln\left(e^{e^{\ln6}}\right)
e
l
n
(
l
n
e
7
)
+
ln
(
e
e
l
n
6
)
Logarithmic Functions
Simplify
log
7
8
log
7
32
\frac{\log_78}{\log_732}
l
o
g
7
32
l
o
g
7
8
Logarithmic Functions
Evaluate
e
[
2
ln
4
−
ln
2
]
+
ln
(
e
3
e
2
)
e^{[2\ln4-\ln2]}+\ln(e^3e^2)
e
[
2
l
n
4
−
l
n
2
]
+
ln
(
e
3
e
2
)
.
Logarithmic Functions
Find the inverse of
f
(
x
)
=
ln
(
x
−
1
x
+
1
)
f\left(x\right)=\ln\left(\frac{x-1}{x+1}\right)
f
(
x
)
=
ln
(
x
+
1
x
−
1
)
Logarithmic Functions
Find the domain of the following function:
f
(
x
)
=
x
2
−
2
x
+
1
ln
(
x
2
−
5
x
+
6
)
f(x)=\dfrac{x^2-2x+1}{\sqrt{\ln(x^2-5x+6)}}
f
(
x
)
=
ln
(
x
2
−
5
x
+
6
)
x
2
−
2
x
+
1
Practice: Inverse function
Find the inverse of
f
(
x
)
=
ln
1
8
−
2
−
3
cos
x
f\left(x\right)=\ln\frac{1}{\sqrt{8-2^{-3\cos x}}}
f
(
x
)
=
ln
8
−
2
−
3
c
o
s
x
1
,
0
≤
x
<
π
0 \leq x < \pi
0
≤
x
<
π
.
Logarithmic Functions
Solve for
x
x
x
in the equation
ln
(
ln
(
x
+
1
)
)
=
0
\ln\left(\ln\left(x+1\right)\right)=0
ln
(
ln
(
x
+
1
)
)
=
0
Logarithmic Functions
Solve for
x
x
x
in the equation
ln
(
3
x
−
1
)
=
−
e
\ln\left(3x-1\right)=-e
ln
(
3
x
−
1
)
=
−
e
.
Logarithmic Functions
Find the inverse function of
f
(
x
)
=
2
ln
(
x
+
e
)
f\left(x\right)=2\ln\left(x+e\right)\
f
(
x
)
=
2
ln
(
x
+
e
)
.
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Solve for
x
x
x
in the equation
ln
(
ln
(
x
+
1
)
)
=
0
\ln\left(\ln\left(x+1\right)\right)=0
ln
(
ln
(
x
+
1
)
)
=
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
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
a.
f
−
1
(
x
)
=
ln
(
3
x
−
1
)
−
2
f^{-1}\left(x\right)=\ln\left(3x-1\right)-2
f
−
1
(
x
)
=
ln
(
3
x
−
1
)
−
2
; domain of
f
−
1
f^{-1}
f
−
1
is
(
1
3
,
∞
)
\left(\frac{1}{3},\infty\right)
(
3
1
,
∞
)
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
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Practice: Solving a logarithmic function
Practice Question: Solving a Logarithmic Function
Solve for 𝑥 in the equation
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
2\ln\left(x-1\right)-\ln\left(4\right)=0
2
ln
(
x
−
1
)
−
ln
(
4
)
=
0
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥
2
− 5) on its appropriate domain.
Logarithmic Functions
Find the domain of the function
f
(
x
)
=
ln
(
x
2
−
5
)
f\left(x\right)=\ln\left(x^2-5\right)
f
(
x
)
=
ln
(
x
2
−
5
)
.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse
f
−
1
f^{-1}
f
−
1
of the function
f
(
x
)
=
2
−
e
3
x
+
1
f\left(x\right)=2-e^{3x+1}
f
(
x
)
=
2
−
e
3
x
+
1
, and determine its domain.
Domain of Log Functions
Find the domain of the function
f
(
x
)
=
ln
(
ln
(
x
2
)
)
f\left(x\right)=\ln\left(\ln\left(x^2\right)\right)
f
(
x
)
=
ln
(
ln
(
x
2
)
)
.
Logarithmic Functions: Domain and Range
Find the formula of the function obtained by shifting the graph of
y
=
ln
x
y=\ln x
y
=
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
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Logarithmic Functions: Log Values
Practice: Log Values
If
x
=
log
2
1
8
x=\log_2\sqrt{\frac{1}{8}}
x
=
lo
g
2
8
1
,
y
=
ln
(
10
e
2
)
+
ln
(
0.1
)
y=\ln\left(\frac{10}{e^2}\right)+\ln\left(0.1\right)
y
=
ln
(
e
2
10
)
+
ln
(
0.1
)
, and
z
=
e
−
2
ln
3
z=e^{-2\ln3}
z
=
e
−
2
l
n
3
, which of the following statements is true?
Solving Log Equations
If
e
⋅
ln
(
1
+
e
x
)
=
1
e\cdot\ln\left(1+e^x\right)=1
e
⋅
ln
(
1
+
e
x
)
=
1
, solve for
x
x
x