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Precalculus : Exponential and Logarithmic Functions

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Example Questions

Example Question #24 : Solve Logarithmic Equations

Condense the logarithm

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

Possible Answers:

$3log\,xy$

$log\,3xy$

$log\,(xy)^3$

$log\,x^3y$

Correct answer:

$log\,x^3y$

Explanation:

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

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

As such

$3log\,x+log\,y=log\,x^3+log\,y=log\,x^3y$

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Example Question #22 : Solve Logarithmic Equations

Expand the following logarithmic expression:

$\log \left(\frac{8x^2y}{z^3}\right)$

Possible Answers:

$\log (8)+\frac{1}{2}\log (x)+\log (y)-\frac{1}{3}\log (z)$

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

$\log (8)-\log (x^2)-\log (y)+\log (z^3)$

$\log (8)+2\log (x)+\log (y)-3\log (z)$

$\log (8x^2y)+\log (z^3)$

Correct answer:

$\log (8)+2\log (x)+\log (y)-3\log (z)$

Explanation:

We start expanding our logarithm by using the following property:

$\log \left(\frac{a}{b}\right)=\log(a)-\log (b)$

$\log \left(\frac{8x^2y}{z^3}\right)=\log (8x^2y)-\log (z^3)$

Now we have two terms, and we can further expand the first term with the following property:

$\log (abc)=\log (a)+\log (b)+\log (c)$

$\log (8x^2y)=\log (8)+\log (x^2)+\log (y)$

$\log \left(\frac{8x^2y}{z^3}\right)=\log (8)+\log (x^2)+\log (y)-\log (z^3)$

Now we only have two logarithms left with nonlinear terms, which we can expand using one final property:

$\log (a^b)=b\log (a)$

Using this property on our two terms with exponents, we obtain the final expanded expression:

$\log \left(\frac{8x^2y}{z^3}\right)=\log (8)+2\log (x)+\log (y)-3\log (z)$

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Example Question #23 : Solve Logarithmic Equations

Expand

$\log_{2}\left(\frac{2}{9x+2}\right)$ .

Possible Answers:

$-log_{2}(9x+2)$

$-log_{2}(9x)$

$-log_{2}(9x)+1$

3+x

$1-log_{2}(9x+2)$

Correct answer:

$1-log_{2}(9x+2)$

Explanation:

To expand

$\log_{2}\left(\frac{2}{9x+2}\right)$ , use the quotient property of logs.

The quotient property states:

$\log_{b} (xy)=\log_{b}x-\log_{b}y$

Substituting in our given information we get:

$\log_{2}\left(\frac{2}{9x+2}\right)= \log_{2}2-\log_{2}(9x+2) = 1-\log_{2}(9x+2)$

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Example Question #1186 : Pre Calculus

Which of the following correctly expresses the following logarithm in expanded form?

$log\left(\frac{5x^2}{y^3}\right)$

Possible Answers:

$\frac{5log(x^2)}{log(y^3)}$

log(5)+log^2(x)-log^3(y)

log(5)+2log(x)-3log(y)

log(5)-2log(x)+3log(y)

2log(5)+2log(x)-3log(y)

Correct answer:

log(5)+2log(x)-3log(y)

Explanation:

Begin by recalling a few logarithm rules:

1) When adding logarithms of like base, multiply the inside.

2) When subtracting logarithms of like base, subtract the inside.

3) When multiplying a logarithm by some number, raise the inside to that power.

Keep these rules in mind as we work backward to solve this problem:

$log\left(\frac{5x^2}{y^3}\right)$

Using rule 2), we can get the following:

log(5x^2)-log(y^3)

Next, use rule 1) on the first part to get:

log(5)+log(x^2)-log(y^3)

Finally, use rule 3) on the second and third parts to get our final answer:

log(5)+2log(x)-3log(y)

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Example Question #1187 : Pre Calculus

Express $log\left(\frac{xy^2}{h}\right)$ in its expanded, simplified form.

Possible Answers:

$\frac{log(x)+2log(y)}{log(h)}$

log(xy^2)-log(h)

log(x)+log(2y)-log(h)

log(x)+2log(y)-log(h)

Correct answer:

log(x)+2log(y)-log(h)

Explanation:

Using the properties of logarithms, expand the logrithm one step at a time:

When expanding logarithms, division becomes subtration, multiplication becomes division, and exponents become coefficients.

$log\left(\frac{xy^2}{h}\right)=log(xy^2)-log(h)$ .

=log(x)+log(y^2)-log(h)

=log(x)+2log(y)-log(h)

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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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Precalculus : Exponential and Logarithmic Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #24 : Solve Logarithmic Equations

Example Question #22 : Solve Logarithmic Equations

Example Question #23 : Solve Logarithmic Equations

Example Question #1186 : Pre Calculus

Example Question #1187 : Pre Calculus

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

All Precalculus Resources

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