Precalculus Examples

$- \frac{2}{3} \cdot \log_{3} (2 m^{2}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1

Simplify each term.

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Step 1.1

Multiply $- \frac{2}{3} \log_{3} (2 m^{2})$ .

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Step 1.1.1

Reorder $\log_{3} (2 m^{2})$ and $\frac{2}{3}$ .

$- (\frac{2}{3} \log_{3} (2 m^{2})) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.1.2

Simplify $\frac{2}{3} \log_{3} (2 m^{2})$ by moving $\frac{2}{3}$ inside the logarithm.

$- \log_{3} ({(2 m^{2})}^{\frac{2}{3}}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.2

Apply the product rule to $2 m^{2}$ .

$- \log_{3} (2^{\frac{2}{3}} {(m^{2})}^{\frac{2}{3}}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.3

Multiply the exponents in ${(m^{2})}^{\frac{2}{3}}$ .

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Step 1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{2 (\frac{2}{3})}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.3.2

Multiply $2 (\frac{2}{3})$ .

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Step 1.3.2.1

Combine $2$ and $\frac{2}{3}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{2 \cdot 2}{3}}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.3.2.2

Multiply $2$ by $2$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \frac{1}{2} \cdot \log_{3} (4 m^{2})$

Step 1.4

Simplify $\frac{1}{2} \log_{3} (4 m^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} ({(4 m^{2})}^{\frac{1}{2}})$

Step 1.5

Apply the product rule to $4 m^{2}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (4^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 1.6

Rewrite $4$ as $2^{2}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} ({(2^{2})}^{\frac{1}{2}} {(m^{2})}^{\frac{1}{2}})$

Step 1.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 1.8

Cancel the common factor of $2$ .

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Step 1.8.1

Cancel the common factor.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2^{2 (\frac{1}{2})} {(m^{2})}^{\frac{1}{2}})$

Step 1.8.2

Rewrite the expression.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2^{1} {(m^{2})}^{\frac{1}{2}})$

Step 1.9

Evaluate the exponent.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2 {(m^{2})}^{\frac{1}{2}})$

Step 1.10

Multiply the exponents in ${(m^{2})}^{\frac{1}{2}}$ .

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Step 1.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2 m^{2 (\frac{1}{2})})$

Step 1.10.2

Cancel the common factor of $2$ .

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Step 1.10.2.1

Cancel the common factor.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2 m^{2 (\frac{1}{2})})$

Step 1.10.2.2

Rewrite the expression.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2 m^{1})$

Step 1.11

Simplify.

$- \log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}}) + \log_{3} (2 m)$

Step 2

Simplify each term.

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Step 2.1

Rewrite $\log_{3} (2^{\frac{2}{3}} m^{\frac{4}{3}})$ using the change of base formula.

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Step 2.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{3} (2 m)$

Step 2.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$- \frac{\log (2^{\frac{2}{3}} m^{\frac{4}{3}})}{\log (3)} + \log_{3} (2 m)$

Step 2.2

Rewrite $\log_{3} (2 m)$ using the change of base formula.

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Step 2.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- \frac{\log (2^{\frac{2}{3}} m^{\frac{4}{3}})}{\log (3)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$- \frac{\log (2^{\frac{2}{3}} m^{\frac{4}{3}})}{\log (3)} + \frac{\log (2 m)}{\log (3)}$