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Precalculus : Exponential and Logarithmic Functions

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Example Questions

Example Question #62 : Properties Of Logarithms

Express the following in expanded form.

$\log{\frac{xz}{y}}$

Possible Answers:

$\log{x}+\log{z}-\log{y}$

$\log{x}-\log{z}+\log{y}$

$\log{x}+\log{z}+\log{y}$

$\log{x}-\log{z}-\log{y}$

Correct answer:

$\log{x}+\log{z}-\log{y}$

Explanation:

To solve, simply remember that when adding logs, you multiply their insides and when subtract logs, you divide your insides. You must use this in reverse to solve. Thus,

$\log{\frac{xz}{y}}=\log{xz}-\log{y}=\log{x}+\log{z}-\log{y}$

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Example Question #37 : Solve Logarithmic Equations

Completely expand this logarithm:

$\ln \frac{\sqrt{6x-2}}{8}$

Possible Answers:

The answer is not present.

$\ln(6x-2)^{\frac{1}{2}}-\ln 8$

$\frac{1}{2}\ln(6x-2)-\ln 8$

$\frac{1}{2}\ln(6x) -\ln (2)-\ln 8$

$\ln(6x-2)^{\frac{1}{2}}-2.08$

Correct answer:

$\frac{1}{2}\ln(6x-2)-\ln 8$

Explanation:

We expand logarithms using the same rules that we use to condense them.

Here we will use the quotient property

$\ln\frac{u}{v}= \ln u-\ln v$

and the power property

$\ln u^{n}=n\ln u$ .

$\ln \frac{\sqrt{6x-2}}{8}$

Use the quotient property:

$\ln \sqrt{6x-2}-\ln 8$

Rewrite the radical:

$\ln (6x-2)^{\frac{1}{2}}-\ln 8$

Now use the power property:

$\frac{1}{2}\ln (6x-2)-\ln 8$

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Example Question #63 : Properties Of Logarithms

Expand the logarithm

$log\,(x^2y^3)$

Possible Answers:

$2log\,x+3log\,y$

$6log\,xy$

$log\,2x+log\,3y$

$log\,x^2-log\,y^3$

Correct answer:

$2log\,x+3log\,y$

Explanation:

In order to expand the logarithmic expression, we use the following properties

$alog\,x=log\,x^a$
$log\,a+log\,b=log\,ab$

As such

$log\,(x^2y^3)=log\,x^2+log\,y^3=2log\,x+3log\,y$

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Example Question #64 : Properties Of Logarithms

Given the equation $ln(e^{3x})=15$ , what is the value of ? Use the inverse property to aid in solving.

Possible Answers:

x=3

x=ln3

x=3e

$x=e^{15}$

x=5

Correct answer:

x=5

Explanation:

The natural logarithm and natural exponent are inverses of each other. Taking the of will simply result in the argument of the exponent.

That is $ln(e^{3x})=3x$

Now, 3x=15 , so x=5.

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Example Question #1 : Graph Logarithms

What is the domain of the function f(x)=2ln(x+2)-1

Possible Answers:

$[-2,\infty ]$

$(-2,\infty )$

$(0,\infty ]$

$[0,\infty ]$

$[-2,\infty )$

Correct answer:

$(-2,\infty )$

Explanation:

The function ln(x) is undefined unless x>0 . Thus ln(x+2) is undefined unless x>-2 because the function has been shifted left.

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Example Question #2 : Graph Logarithms

What is the range of the function f(x)=2ln(x+2)-1

Possible Answers:

$(-\infty,\infty]$

$[-\infty,\infty]$

$[0,\infty)$

$[-\infty,\infty)$

$[-10.21,\infty)$

Correct answer:

$[-10.21,\infty)$

Explanation:

To find the range of this particular function we need to first identify the domain. Since $ln(x)>0 \rightarrow ln(x+2)>-2$ we know that x=-2 is a bound on our function.

From here we want to find the function value as approaches .

To find this approximate value we will plug in -1.99 into our original function.

f(-1.99)=2ln(-1.99+2)-1

=2(-4.61)-1

=-9.21-1

=-10.21

This is our lowest value we will obtain. As we plug in large values we get large function values.

Therefore our range is:

$[-10.21,\infty)$

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Example Question #3 : Graph Logarithms

Which of the following logarithmic functions match the provided diagram?

Varsity log graph

Possible Answers:

$y=log_{0.75}x$

$y=log_{0.1}x$

$y=log_{0.5}x$

$y=log_{10}x$

$y=log_{1}x$

Correct answer:

$y=log_{0.5}x$

Explanation:

Looking at the diagram, we can see that when y=1 , x=0.5 . Since represents the exponent and represents the product, and any base with an exponent of 1 equals the base, we can determine the base to be 0.5.

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Example Question #4 : Graph Logarithms

Which of the following diagrams represents the graph of the following logarithmic function?

$y=log_{5}x$

Possible Answers:

Varsity log graph

Correct answer:

Varsity log graph

Explanation:

For $y=log_{5}x$ , is the exponent of base 5 and is the product. Therefore, when y=1 , $x=5^{1}$ and when y=3 , $x=5^{3 }$ . As a result, the correct graph will have values of 5 and 125 at y=1 and y=3 , respectively.

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Example Question #4 : Graph Logarithms

Which of the following diagrams matches the given logarithmic function:

$x=log_{2}y$

Possible Answers:

Varsity log graph

Correct answer:

Varsity log graph

Explanation:

For this function, represents the exponent and y represents the product of the base 2 and its exponent. On the diagram, it is clear that as the value increases, the value increases exponentially and at x=1 , y=2 . Those two characteristics of the graph indicate that x is the exponent value and the base is equal to 2.

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Example Question #5 : Graph Logarithms

Which of the following logarithmic functions match the given diagram?

Varsity log graph

Possible Answers:

$y=log_{3}x$

$x=log_{2}y$

$y=log_{0.5}x$

$x=log_{3}y$

$y=log_{2}x$

Correct answer:

$x=log_{3}y$

Explanation:

Looking at the graph, the y-value diminishes exponentially as decreases and increases rapidly as the x-value increases, which indicates that is the exponent value for the equation.

Also, y=3 when x=1 and y=27 when x=3 , which can be expressed as $3^{1}=3$ and $3^{3}=27$ , respectively.

This indicates that the diagram is consistent with the function $x=log_{3}y$ .

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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #62 : Properties Of Logarithms

Example Question #37 : Solve Logarithmic Equations

Example Question #63 : Properties Of Logarithms

Example Question #64 : Properties Of Logarithms

Example Question #1 : Graph Logarithms

Example Question #2 : Graph Logarithms

Example Question #3 : Graph Logarithms

Example Question #4 : Graph Logarithms

Example Question #4 : Graph Logarithms

Example Question #5 : Graph Logarithms

All Precalculus Resources

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