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Precalculus : Exponential and Logarithmic Functions

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Example Questions

Example Question #2 : Graph Exponential Functions

Choose the description that matches the equation below:

$y = (\frac{1}{3})^{4x}$

Possible Answers:

Exponential decay

y = 1

Exponential decay

$y = \frac{1}{3}$

Exponential decay

y = 0

Exponential growth

y = 1

Exponential growth

y = 3

Correct answer:

Exponential decay

y = 1

Explanation:

Exponential graphs can either decay or grow. This is based on the value of the base of the exponent. If the base is greater than , the graph will be growth. And, if the base is less than , then the graph will be decay. In this situation, our base is $\frac{1}{3}$ . Since this is less than , we have a decay graph. Then, to determine the y-intercept we substitute x=0 . Thus, we get:

$y = (\frac{1}{3})^{(4\cdot 0)} = (\frac{1}{3})^0=1$ for the y-intercept.

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Example Question #131 : Exponential And Logarithmic Functions

Which of the following represents the graph of -(2^x) ?

Possible Answers:

Graph1

Graph4

Graph3

Graph2

Correct answer:

Graph1

Explanation:

Note that the negative sign in this function comes outside of the parentheses. This should show you that the bigger the number in parentheses, the lower the curve of the graph will go. Since this is an exponential function, the larger that the x value gets, then, the "more negative" this graph will go. The graph closest to zero on the left-hand side - where x is negative - and then shoots down and to the right rapidly when x gets larger is the correct graph.

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Example Question #2 : Graphing

Define a function as follows:

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

Give the -intercept of the graph of .

Possible Answers:

$\left ( 0, \frac{3}{4}\right )$

(0,13)

(0,2)

(0,1)

(0,-3)

Correct answer:

(0,13)

Explanation:

The -coordinate ofthe -intercept of the graph of is 0, and its -coordinate is f(0) :

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

$f(0) = 4 ^{0+2} - 3 = 4 ^{2} - 3 = 16 - 3 = 13$

The -intercept is the point (0,13) .

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Example Question #122 : Solving And Graphing Exponential Equations

Does the function $y = 4^{x}$ have any -intercepts?

Possible Answers:

Yes, x = 1

Yes, x = 4 and x = -4

Yes, x = 0

That cannot be determined from the information given.

Correct answer:

Explanation:

The -intercept of a function is where y = 0 . Thus, we are looking for the -value which makes $0 = 4^{x}$ .

If we try to solve this equation for we get an error.

To bring the exponent down we will need to take the natural log of both sides.

ln(0)=xln(4)

Since the natural log of zero does not exist, there is no exponent which makes this equation true.

Thus, there is no -intercept for this function.

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Example Question #21 : Solving Exponential Functions

Which of the following functions represents exponential decay?

Possible Answers:

$y = 5^{x}$

$y = 3(2^{x})$

$y = \left(\frac{1}{3}\right)^{x}$

$y = 6^{x + 3}$

$y = (-2)^{x}$

Correct answer:

$y = \left(\frac{1}{3}\right)^{x}$

Explanation:

Exponential decay describes a function that decreases by a factor every time increases by .

These can be recognizable by those functions with a base which is between and .

The general equation for exponential decay is,

A=A_0b^t where the base is represented by and 0<b<1 .

Thus, we are looking for a fractional base.

The only function that has a fractional base is,

$y = \left(\frac{1}{3}\right)^{x}$ .

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Example Question #2 : Graph Exponential Functions

What is the -intercept of $y = 4(2^{x})$ ?

Possible Answers:

There is no -intercept.

Correct answer:

Explanation:

The -intercept of a graph is the point on the graph where the -value is .

Thus, to find the -intercept, substitute x = 0 and solve for .

Thus, we get:

$y = 4(2^{0}) = 4(1)=4$

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Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solving an exponential equation.

Solve for ,

$5^{x}=250$ .

Possible Answers:

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

x=5

x=3+log2

x=30

$x=3+log_{5}2$

Correct answer:

$x=3+log_{5}2$

Explanation:

We recall the property:

$log_b(x)=y \leftrightarrow b^y=x$

b=5, y=250

log_5(250)=x

$x=log_{5}(125\cdot 2)=log_{5}125+log_{5}2$

Now, $log_{5}125=log_{5}(5^{3})=3$ .

Thus

$x=3+log_{5}2$ .

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Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Solving an exponential equation.

Solve

$10^{3x}=100$

Possible Answers:

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

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

x=3

x=2

x=10

Correct answer:

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

Explanation:

Use log (which is just $log_{10}$ , by convention) to solve.

$log(10^{3x})=log(100)$

$log(10^{3x})=log(10^2)$

3x=2

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

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Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve the equation for using the rules of logarithms.

$\ln(17x)+\ln\left(\frac{x}{3}\right)+\log_3(x^2)=4$

Possible Answers:

$x\approx 1.04$

$x\approx 75.15$

x=3

$x\approx1.86$

x=9

Correct answer:

$x\approx 1.04$

Explanation:

Expanding the logarithms into sums of logarithms will cancel out the first two x terms, resulting in the equation:

$\ln(17)+ln(3)+2\log_3(x)=4$

Combining the first and second terms, then subtracting the new term over will allow you to isolate the variable term.

Divide both sides of the equation by 2, then exponentiate with 3.

Evaluating this term numerically will give the correct answer.

$2\log_3(x)=4-\ln(51)\implies \log_3(x)=2-\frac{\ln(51)}{2}\\ \\ \implies x=3^{2-\ln(51)/2}\approx1.04$

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Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Solve the following equation:

10^x=175

Possible Answers:

$x\approx22.4$

$x\approx2.24$

x=2.24

$x\approx4.48$

$x\approx0.224$

Correct answer:

$x\approx2.24$

Explanation:

To solve this equation, recall the following property:

log_by=x Can be rewritten as b^x=y

10^x=175

$log_{10}175=x$

Evaluate with your calculator to get

$x\approx2.24$

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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #2 : Graph Exponential Functions

Example Question #131 : Exponential And Logarithmic Functions

Example Question #2 : Graphing

Example Question #122 : Solving And Graphing Exponential Equations

Example Question #21 : Solving Exponential Functions

Example Question #2 : Graph Exponential Functions

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #1 : Use Logarithms To Solve Exponential Equations And Inequalities

Example Question #2 : Use Logarithms To Solve Exponential Equations And Inequalities

All Precalculus Resources

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