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Precalculus : Properties of Logarithms

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Example Questions

Example Question #21 : Solve Logarithmic Equations

Completely condense the logarithm: $\frac{1}{3}log_{10}x-\frac{1}{2}[log_{10}(x+y)]$ .

Possible Answers:

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{xy}}$

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}$

None of the other answers.

$log_{10}\frac{\sqrt{x}}{\sqrt{x+y}}$

$log_{10}\frac{\sqrt{x}}{\sqrt{xy}}$

Correct answer:

$log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}$

Explanation:

$\frac{1}{3}log_{10}x-\frac{1}{2}[log_{10}(x+y)]$

Apply the Power property:

$log_{10}x^\frac{1}{3}-log_{10}(x+y)^\frac{1}{2}$

$log_{10}\sqrt[3]{x}-log_{10}\sqrt{x+y}$

Apply the quotient property:

$\mathbf{log_{10}\frac{\sqrt[3]{x}}{\sqrt{x+y}}}$

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Example Question #21 : Solve Logarithmic Equations

Simplify

$\small \log \,3\,\,+\,\,\log\,7$

Possible Answers:

$\small \log\,\frac{3}{7}$

$\small \log\,37$

$\small \log\,21$

$\small \log\,10$

Correct answer:

$\small \log\,21$

Explanation:

By the property of the addition of logarithms with the same base

$\small \small \log\,a\,\,+\,\,\log\,b=\log\,ab$

As such

$\small \small \log \,3\,\,+\,\,\log\,7=\log\,3\cdot7=\log\,21$

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Example Question #23 : Solve Logarithmic Equations

Condense the following equation:

log(x)+4log(y)+log(3)-log(z)

Possible Answers:

$log(\frac{3xy^4}{z})$

Cannot simplify

$log(\frac{12xy}{z})$

log(12xyz)

log(x+4y+3-z)

Correct answer:

$log(\frac{3xy^4}{z})$

Explanation:

Let's use the properties of logarithms to condense this equation. We will use the follwing three properties

Power Rule $b\cdot log(x)=log(x^b)$

Product Rule log(x)+log(y)=log(xy)

Quotient Rule $log(x)-log(y)=log(\frac{x}y{})$

Let's first use the power rule to rewrite the second term

log(x)+log(y^4)+log(3)-log(z)

Then, we'll use the product rule to combine the first threeterms

log(3xy^4)-log(z)

Lastly, we'll use the quotient rule to combine into one term

$log(\frac{3xy^4}{z})$

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Example Question #22 : Solve Logarithmic Equations

Express the following in condensed form:

$\log(x-4)+\log(x)-\log{2}$

Possible Answers:

$\log\frac{x^2-4x}{2}$

$\log(x-4)+\log(x)-\log{2}$

$\log(x-4)+2$

$\log(x-4)(x)(2)$

Correct answer:

$\log\frac{x^2-4x}{2}$

Explanation:

To solve, simply remember that when you add logs, you can multiply their insides and when you subtract them, you can divide.

Thus,

$\\ \log(x-4)+\log(x)-\log{2}\\ \\=\log\frac{(x-4)x}{2}\\ \\=\log\frac{x^2-4x}{2}$

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Example Question #23 : Solve Logarithmic Equations

Express the following in condensed form:

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

Possible Answers:

Same

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

$\log(xy^4)$

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

Correct answer:

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

Explanation:

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

$\log{x^2}+\log{y^2}=\log(x^2y^2)$

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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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Precalculus : Properties of Logarithms

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #21 : Solve Logarithmic Equations

Example Question #21 : Solve Logarithmic Equations

Example Question #23 : Solve Logarithmic Equations

Example Question #22 : Solve Logarithmic Equations

Example Question #23 : Solve Logarithmic Equations

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

All Precalculus Resources

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