Precalculus Examples

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

Step 1

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

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Step 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 1.2

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 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 3

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

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Step 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 3.2

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

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Step 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 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 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 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 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 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 8

Cancel the common factor of $2$ .

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Step 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 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 9

Evaluate the exponent.

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

Step 10

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

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Step 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 10.2

Cancel the common factor of $2$ .

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Step 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 10.2.2

Rewrite the expression.

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

Step 11

Simplify.

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