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College Algebra : Exponential and Logarithmic Functions

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Example Questions

Example Question #1 : Exponential Functions

Solve: e^l^n^(^2^x^+^3^)

Possible Answers:

The answer does not exist.

2x+3

$\frac{1}{2x+3}$

Correct answer:

2x+3

Explanation:

To solve e^l^n^(^2^x^+^3^) , it is necessary to know the property of .

e^l^n^(^x^)=x

Since and the terms cancel due to inverse operations, the answer is what's left of the term.

The answer is: 2x+3

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Example Question #41 : Logarithms

Which equation is equivalent to:

$\log _{4}\left( \frac{1}{2}\right)=x$

Possible Answers:

$(4)^{x}=\left(\frac{1}{2}\right)^{x}$

$4^{\frac{1}{2}}=x$

$\frac{1^{4}}{2}=x^{4}$

$4^{x}=\frac{1}{2}$

$\left(\frac{1}{2}\right)^{4}=x$

Correct answer:

$4^{x}=\frac{1}{2}$

Explanation:

$log{_{a}}^{b}=x$ , b>0

$\Rightarrow a^{x}=b$

So, ${{log_{4}}^{\frac{1}{2}}}=x\Rightarrow 4^{x}=\frac{1}{2}$

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Example Question #3 : Exponential And Logarithmic Functions

$Log_{b}x=y$

What is the inverse of the log function?

Possible Answers:

$y^{b}=x$

$x^{b}=y$

$b=x\cdot y$

$b^{x}=y$

$b^{y}=x$

Correct answer:

$b^{y}=x$

Explanation:

This is a general formula that you should memorize. The inverse of $Log_{b}x=y$ is $b^{y}=x$ . You can use this formula to change an equation from a log function to an exponential function.

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Example Question #4 : Exponential And Logarithmic Functions

Rewrite the following expression as an exponential expression:

log_3b=14,000

Possible Answers:

b^3=14,000

$3^{14,000}=b$

14,000^3=b

$b^{14,000}=3$

Correct answer:

$3^{14,000}=b$

Explanation:

Rewrite the following expression as an exponential expression:

log_3b=14,000

Recall the following property of logs and exponents:

log_b(a)=c

Can be rewritten in the following form:

b^c=a

So, taking the log we are given;

log_3b=14,000

We can rewrite it in the form:

$3^{14,000}=b$

So b must be a really huge number!

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Example Question #5 : Exponential And Logarithmic Functions

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Possible Answers:

147^b=13

$b^{13}=147$

$13^{b}=147$

$13^{147}=b$

Correct answer:

$13^{147}=b$

Explanation:

Convert the following logarithmic equation to an exponential equation:

$log_{13}b=147$

Recall the following:

This

log_b(a)=c

Can be rewritten as

b^c=a

So, our given logarithm

$log_{13}b=147$

Can be rewritten as

$13^{147}=b$

Fortunately we don't need to expand, because this woud be a very large number!

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Example Question #6 : Exponential And Logarithmic Functions

Convert the following logarithmic equation to an exponential equation.

log_3(14x)=156

Possible Answers:

$(14x)^{156}=3$

156=(14x)^3

$3^{14x}=156$

$3^{156}=14x$

Correct answer:

$3^{156}=14x$

Explanation:

Convert the following logarithmic equation to an exponential equation.

log_3(14x)=156

To convert from logarithms to exponents, recall the following property:

log_b(x)=a

Can be rewritten as:

b^a=x

So, starting with

log_3(14x)=156 ,

We can get

$3^{156}=14x$

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Example Question #2 : Exponential Functions

Solve the following:

5^x=125

Possible Answers:

625

Correct answer:

Explanation:

To solve the following, you must "undo" the 5 with taking log based 5 of both sides. Thus,

5^x=125

$\log_5{5^x}=\log_5{125}$

$x=\log_5{125}$

The right hand side can be simplified further, as 125 is a power of 5. Thus,

$x=\log_5{125}$

$x=\log_5{5^3}$

x=3

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Example Question #5 : Exponential And Logarithmic Functions

Solve for :

$2^{x}+ 2 ^{x+2} = 171$

(Nearest hundredth)

Possible Answers:

$x \approx 2.84$

$x \approx 2.71$

$x \approx 3.65$

$x \approx 5.10$

The equation has no solution.

Correct answer:

$x \approx 5.10$

Explanation:

Apply the Product of Powers Property to rewrite the second expression:

$2^{x}+ 2 ^{x+2} = 171$

$2^{x}+ 2 ^{2} \cdot 2 ^{x} = 171$

$1 \cdot 2^{x}+ 4 \cdot 2 ^{x} = 171$

Distribute out:

$(1 + 4) \cdot 2 ^{x} = 171$

$5 \cdot 2 ^{x} = 171$

Divide both sides by 5:

$\frac{5 \cdot 2 ^{x}}{5} = \frac{171}{5}$

$2 ^{x} = 34.2$

Take the natural logarithm of both sides (and note that you can use common logarithms as well):

$\ln 2 ^{x} = \ln 34.2$

Apply a property of logarithms:

$x \ln 2 = \ln 34.2$

Divide by $\ln 2$ and evaluate:

$\frac{x \ln 2}{\ln2} = \frac{\ln 34.2}{\ln2}$

$x = \frac{3.5322 }{0.6931 }$

$x \approx 5.10$

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Example Question #7 : Exponential And Logarithmic Functions

Solve for :

$3^{2x+ 1} = 9^{2x- 4 }$

(Nearest hundredth, if applicable).

Possible Answers:

x= 3.25

x = 4.50

x = 3.40

The equation has no solution.

x= 1.10

Correct answer:

x = 4.50

Explanation:

$9 = 3^{2}$ , so rewrite the expression at right as a power of 3 using the Power of a Power Property:

$3^{2x+ 1} = 9^{2x- 4 }$

$3^{2x+ 1} = (3^{2} )^{2x- 4 }$

$3^{2x+ 1} = 3^{2(2x- 4 )}$

Set the exponents equal to each other and solve the resulting linear equation:

2x+ 1 = 2(2x-4)

Distribute:

2x+ 1 = 4x-8

Subtract and 1 from both sides; we can do this simultaneously:

2x+ 1 - 4x - 1 = 4x-8 - 4x - 1

- 2x = -9

Divide by :

$\frac{- 2x}{-2} = \frac{-9}{-2}$

x =4.5

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Example Question #8 : Exponential And Logarithmic Functions

Solve the following for x:

$\log(2x+4)=1$

Possible Answers:

$\frac{-3}{2}$

Correct answer:

Explanation:

To solve, you must first "undo" the log. Since no base is specified, you assume it is 10. Thus, we need to take 10 to both sides.

$10^{\log(2x+4)}=10^1$

2x+4=10

Now, simply solve for x.

2x=6

x=3

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College Algebra : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #1 : Exponential Functions

Example Question #41 : Logarithms

Example Question #3 : Exponential And Logarithmic Functions

Example Question #4 : Exponential And Logarithmic Functions

Example Question #5 : Exponential And Logarithmic Functions

Example Question #6 : Exponential And Logarithmic Functions

Example Question #2 : Exponential Functions

Example Question #5 : Exponential And Logarithmic Functions

Example Question #7 : Exponential And Logarithmic Functions

Example Question #8 : Exponential And Logarithmic Functions

All College Algebra Resources

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