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Precalculus : Exponential and Logarithmic Functions

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Example Questions

Example Question #1 : Properties Of Logarithms

Find the inverse function of the following exponential function:

$y = \frac{2}{3}e^{4x+1}$

Possible Answers:

$y = \frac{1}{4}\ln {\frac{3}{2}x}-\frac{1}{4}$

$y = \frac{1}{4}\ln {x}-\frac{1}{4}$

$y = \frac{3}{8}\ln {x}-\frac{1}{4}$

$y = \frac{1}{4}\ln {\frac{2}{3}x}-\frac{1}{4}$

$y = \frac{1}{4}\ln {\frac{3}{2}x}-1$

Correct answer:

$y = \frac{1}{4}\ln {\frac{3}{2}x}-\frac{1}{4}$

Explanation:

Since we are looking for an inverse function, we start by swapping the x and y variables in our original equation.

$x = \frac{2}{3}e^{4y+1}$

Now we have to solve for y. To do this we have to work towards isolating y. First we remove the constant multiplier:

$\frac{3}{2}x = \frac{3}{2}*\frac{2}{3}e^{4y+1}$

$\frac{3}{2}x = e^{4y+1}$

Next we eliminate the base on the right side by taking the natural log of both sides.

$\ln\frac{3}{2}x = \ln e^{4y+1}$

$\ln\frac{3}{2}x = {4y+1}$

Subtract 1 and divide by 4:

$\ln\frac{3}{2}x - 1 = {4y+1} - 1$

$\ln\frac{3}{2}x - 1 = 4y$

$\frac{1}{4}\left ( \ln\frac{3}{2}x - 1 \right ) = 4y*\frac{1}{4}$

$\frac{1}{4}\ln {\frac{3}{2}x}-\frac{1}{4} = y$

$y = \frac{1}{4}\ln {\frac{3}{2}x}-\frac{1}{4}$

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Example Question #11 : Properties Of Logarithms

Condense the following expression into one logarithm:

$\ln{8x}+2\ln3x^2-\ln2$

Possible Answers:

$\ln36x^5$

$\ln12x^3$

$\ln36x^3$

$\ln12x^5$

Correct answer:

$\ln36x^5$

Explanation:

To condense this, the second term must have the 2 in front moved inside.

$\ln{8x}+\ln(3x^2)^2-\ln2$

$\ln8x+\ln9x^4-\ln2$

When adding two logs, multiply their insides; when subtracting two logs, divide their insides.

$\ln((8x)(9x^4))-\ln2$

$\ln72x^5-\ln2$

$\ln\frac{72x^5}{2}$

$\ln36x^5$

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Example Question #21 : Exponential And Logarithmic Functions

Expand the following log completely

$\log_{3}(27x^4y^3)$

Possible Answers:

$3-4\log_{3}x-3\log_{3}y$

$\log_327+\log_3x^4+3\log_3y^3$

$\log_327+4\log_3x+3\log_3y$

$3+4\log_{3}x+3\log_{3}y$

$\log_327+\log_3x+\log_3y$

Correct answer:

$3+4\log_{3}x+3\log_{3}y$

Explanation:

To expand a logarithm, quantities in the inside that are multiplied get added and quantities in the inside that are divided get subtracted.

$\log_{3}27+\log_3(x^4)+\log_3(y^3)$

$\log_33^3+4\log_3(x)+3\log_3(y)$

$3+4\log_3(x)+3\log_3(y)$

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Example Question #22 : Exponential And Logarithmic Functions

Solve for :

$e^{2x}-5e^x+6=0$

Possible Answers:

$x=\ln3$

$x=\ln2$

$x=\ln6$

$x=\ln2$

$x=\ln3$

$x=\ln5$

Correct answer:

$x=\ln2$

$x=\ln3$

Explanation:

The first step to solving this problem is to realize that

$e^{2x}=(e^x)^2$

Then, the equation falls into the follow form which resembles a quadratic.

(e^x)^2-5e^x+6=0

Let y=e^x . Then,

y^2-5y+6=0

(y-3)(y-2)=0

Thus, y=2 and y=3 .

Since y=e^x ,

$2=e^x\Rightarrow \ln2=\ln{e^x}\Rightarrow x=\ln2$

$3=e^x\Rightarrow \ln3=\ln{e^x}\Rightarrow x=\ln3$

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Example Question #23 : Exponential And Logarithmic Functions

Simplify the expression.

$\log(72x)-\log(12)$

Possible Answers:

$\log(6x)$

$\log(7x)$

$\log(-6x)$

None of the above answers

Correct answer:

$\log(6x)$

Explanation:

Using the quotient rule for logarithms we can condense these two logarithms into a single logarithm.

$\log\left(\frac{72x}{12}\right)$

We then obtain our answer by simple division.

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Example Question #24 : Exponential And Logarithmic Functions

Simplify the expression.

$-2\ln(xy^2)+3\ln(y)$

Possible Answers:

$\ln\left(\frac{y}{x^2}\right)$

-6xy^3

$\ln\left(\frac{1}{x^2y}\right)$

$-6\ln(xy^3)$

$\ln(x^2y)$

Correct answer:

$\ln\left(\frac{1}{x^2y}\right)$

Explanation:

Using the properties of logarithms we first simplify the expression to $\ln(y^3)-\ln(x^2y^4)$ . Then we use the quotient rule for logarithms and cancel some terms to obtain our answer.

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Example Question #25 : Exponential And Logarithmic Functions

Simplify the expression.

$2\log(x)-3\log(y)+\log(x^2y^3)$

Possible Answers:

None of the other answers

$-6\log(x^4)$

-6x^3y^4

$\log(x^4)$

$\log(x^4y^6)$

Correct answer:

$\log(x^4)$

Explanation:

Using the properties of logarithms we first rewrite the expression as $\log(x^2)-\log(y^3)+\log(x^2y^3)$ . Now we combine the three pieces into the form $\log\left(\frac{x^2x^2y^3}{y^3}\right)$ . We then obtain our answer when we combine the terms with and cancel those with .

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Example Question #26 : Exponential And Logarithmic Functions

Solve for x:

$4^{x-2}=64$

Possible Answers:

x=5

x=3

x=4

x=2

x=1

Correct answer:

x=5

Explanation:

$4^{x-2}=64$

$4^{x-2}=4^3$

$\log_{4}4^{x-2}=\log_{4}4^3$

$x-2=3\Rightarrow x=5$

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Example Question #24 : Exponential And Logarithmic Functions

Which of the following is equivalent to $(e^{3}\cdot e^{9})^{2}$ ?

Possible Answers:

$e^{14}$

$e^{36}$

$e^{24}$

$e^{54}$

$e^{144}$

Correct answer:

$e^{24}$

Explanation:

When multiplying exponents with a common base, you add both the exponents together. Hence, $(e^{3}\cdot e^{9})=e^{12}$

When an exponent is raised to an exponent you multiply the exponents together. Hence, for $(e^{12})^{2}$ you would multiply $(12\cdot 2)$ to get $e^{24}$ .

Answer: $e^{24}$

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Example Question #27 : Exponential And Logarithmic Functions

Solve the following for x:

$3^{2x+1}=81$

Possible Answers:

$\frac{5}{2}$

$\frac{3}{2}$

Correct answer:

$\frac{3}{2}$

Explanation:

$3^{2x+1}=81$

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

$\log_33^{2x+1}=\log_33^4$

2x+1=4

2x=3

$x=\frac{3}{2}$

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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Properties Of Logarithms

Example Question #11 : Properties Of Logarithms

Example Question #21 : Exponential And Logarithmic Functions

Example Question #22 : Exponential And Logarithmic Functions

Example Question #23 : Exponential And Logarithmic Functions

Example Question #24 : Exponential And Logarithmic Functions

Example Question #25 : Exponential And Logarithmic Functions

Example Question #26 : Exponential And Logarithmic Functions

Example Question #24 : Exponential And Logarithmic Functions

Example Question #27 : Exponential And Logarithmic Functions

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