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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

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12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #33 : Solve Logarithmic Equations

Completely expand this logarithm: $\ln\frac{x^4\sqrt{y}}{n^5}$

Possible Answers:

$4\ln x+\ln y-5\ln n$

$4\ln x-\frac{1}{2}\ln y+5\ln n$

$4\ln x+\frac{1}{2}\ln y+5\ln n$

$4\ln x+\frac{1}{2}\ln y-5\ln n$

$4\ln x+\ln y+5\ln n$

Correct answer:

$4\ln x+\frac{1}{2}\ln y-5\ln n$

Explanation:

$\ln\frac{x^4\sqrt{y}}{n^5}$

Quotient property:

$\ln x^4\sqrt{y}-\ln n^5$

Product property:

$\ln x^4+\ln\sqrt{y}-\ln n^5$

Power property:

$\mathbf{4\ln x+\frac{1}{2} \ln y-5 \ln n}$

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Example Question #31 : Solve Logarithmic Equations

Fully expand: $\log(\frac{a^2b}{c})$

Possible Answers:

$\log(2)+\log(a)+\log(b)-\log(c)$

$\log(a^2)+ \log(b)-\log(c)$

$\log(2a^2b)-\log(c)$

$2\log(a)+\log(b)-\log(c)$

$\log(2ab)-\log(c)$

Correct answer:

$2\log(a)+\log(b)-\log(c)$

Explanation:

In order to expand the expression, use the log rules of multiplication and division. Anytime a variable is multiplied, the log is added. If the variable is being divided, subtract instead.

$\log(\frac{a^2b}{c}) = \log(a^2)+ \log(b)-\log(c)$

When there is a power to a variable when it is inside the log, it can be pulled down in front of the log as a coefficient.

$\log(a^2)+ \log(b)-\log(c) = 2\log(a)+\log(b)-\log(c)$

The answer is: $2\log(a)+\log(b)-\log(c)$

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Example Question #31 : Solve Logarithmic Equations

Expand the following:

$\log(2x)+\log(x-1)$

Possible Answers:

$\log(2x)+\log(x-1)$

$\log(2)+x$

$\log(2x^2-2x)$

$\log\frac{2x}{x-1}$

Correct answer:

$\log(2x^2-2x)$

Explanation:

To solve, simply remember that when you add logs, you multiply their insides.

Thus,

$\\\log(2x)+\log(x-1)\\ \\=\log((2x)(x-1))\\ \\=\log(2x^2-2x)$

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Example Question #32 : Solve Logarithmic Equations

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 #33 : Solve Logarithmic Equations

Expand the logarithm

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

Possible Answers:

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

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

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

$6log\,xy$

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 #34 : Solve Logarithmic Equations

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

Possible Answers:

x=ln3

x=3e

$x=e^{15}$

x=5

x=3

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 #103 : Exponential And Logarithmic Functions

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 #104 : Exponential And Logarithmic Functions

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

Possible Answers:

$[-10.21,\infty)$

$[-\infty,\infty)$

$[-\infty,\infty]$

$(-\infty,\infty]$

$[0,\infty)$

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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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #33 : Solve Logarithmic Equations

Example Question #31 : Solve Logarithmic Equations

Example Question #31 : Solve Logarithmic Equations

Example Question #32 : Solve Logarithmic Equations

Example Question #37 : Solve Logarithmic Equations

Example Question #33 : Solve Logarithmic Equations

Example Question #34 : Solve Logarithmic Equations

Example Question #103 : Exponential And Logarithmic Functions

Example Question #104 : Exponential And Logarithmic Functions

Example Question #3 : Graph Logarithms

All Precalculus Resources

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