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Precalculus : Properties of Logarithms

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Example Questions

Example Question #1184 : Pre Calculus

Expand this logarithm: $\log_{3}\frac{6x^3y^2}{x^2}$

Possible Answers:

None of the other answers.

$\log_{3}6+\log_{3}x^3+\log_{3}y^2-\log_{3}x^2$

$\log_{3}6+3\log_{3}x+2\log_{3}y+2\log_{3}x$

$\log_{3}6+\log_{3}x^3+\log_{3}y^2+\log_{3}x^2$

$\log_{3}6+3\log_{3}x+2\log_{3}y-2\log_{3}x$

Correct answer:

$\log_{3}6+3\log_{3}x+2\log_{3}y-2\log_{3}x$

Explanation:

$\log_{3}\frac{6x^3y^2}{x^2}$

Use the Quotient property of Logarithms to express on a single line:

$\log_{3}6x^3y^2-x^2$

Use the Product property of Logarithms to expand the two terms further:

$\log_{3}6+\log_{3}x^3+\log_{3}y^3-\log_{3}x^2$

Finally use the Power property of Logarithms to remove all exponents:

$\log_{3}6+3\log_{3}x+2\log_{3}y-2\log_{3}x$

The expression is now fully expanded.

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Example Question #32 : Solve Logarithmic Equations

Expand the following logarithm:

$\log(\frac{5x^4}{7})$

Possible Answers:

$\frac{\log(5)\cdot 4\log(x)}{-\log(7)}$

$4\log(5)+4\log(x)-4\log(7)$

$\log(5)+4\log(x)-\log(7)$

$\frac{\log(5)\cdot4\log(x)}{\log(7)}$

Correct answer:

$\log(5)+4\log(x)-\log(7)$

Explanation:

Expand the following logarithm:

$\log(\frac{5x^4}{7})$

To expand this log, we need to keep in mind 3 rules:

1) When dividing within a $\log$ , we need to subtract

2) When multiplying within a $\log$ , we need to add

3) When raising to a power within a $\log$ , we need to multiply by that number

These will make more sense once we start applying them.

First, let's use rule number 1

$\log(\frac{5x^4}{7})=\log(5x^4)-\log(7)$

Next, rule 2 sounds good.

$\log(5x^4)-\log(7)=\log(5)+\log(x^4)-\log(7)$

Finally, use rule 3 to finish up!

$\log(5)+\log(x^4)-\log(7)=\log(5)+4\log(x)-\log(7)$

Making our answer

$\log(5)+4\log(x)-\log(7)$

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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 #61 : Properties Of Logarithms

Expand the following:

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

Possible Answers:

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

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

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

$\log(2)+x$

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 #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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Precalculus : Properties of Logarithms

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1184 : Pre Calculus

Example Question #32 : Solve Logarithmic Equations

Example Question #33 : Solve Logarithmic Equations

Example Question #31 : Solve Logarithmic Equations

Example Question #61 : Properties Of Logarithms

Example Question #62 : Properties Of Logarithms

Example Question #37 : Solve Logarithmic Equations

Example Question #63 : Properties Of Logarithms

Example Question #64 : Properties Of Logarithms

All Precalculus Resources

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