Precalculus Examples

$- \frac{3}{4} \cdot \log_{3} (16 p^{4}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 1

Multiply $- \frac{3}{4} \log_{3} (16 p^{4})$ .

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Step 1.1

Reorder $\log_{3} (16 p^{4})$ and $\frac{3}{4}$ .

$- (\frac{3}{4} \log_{3} (16 p^{4})) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 1.2

Simplify $\frac{3}{4} \log_{3} (16 p^{4})$ by moving $\frac{3}{4}$ inside the logarithm.

$- \log_{3} ({(16 p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 2

Apply the product rule to $16 p^{4}$ .

$- \log_{3} (16^{\frac{3}{4}} {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 3

Rewrite $16$ as $2^{4}$ .

$- \log_{3} ({(2^{4})}^{\frac{3}{4}} {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 4

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

$- \log_{3} (2^{4 (\frac{3}{4})} {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 5

Cancel the common factor of $4$ .

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Step 5.1

Cancel the common factor.

$- \log_{3} (2^{4 (\frac{3}{4})} {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 5.2

Rewrite the expression.

$- \log_{3} (2^{3} {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 6

Raise $2$ to the power of $3$ .

$- \log_{3} (8 {(p^{4})}^{\frac{3}{4}}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 7

Multiply the exponents in ${(p^{4})}^{\frac{3}{4}}$ .

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Step 7.1

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

$- \log_{3} (8 p^{4 (\frac{3}{4})}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 7.2

Cancel the common factor of $4$ .

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Step 7.2.1

Cancel the common factor.

$- \log_{3} (8 p^{4 (\frac{3}{4})}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 7.2.2

Rewrite the expression.

$- \log_{3} (8 p^{3}) - \frac{2}{3} \cdot \log_{3} (8 p^{3})$

Step 8

Multiply $- \frac{2}{3} \log_{3} (8 p^{3})$ .

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Step 8.1

Reorder $\log_{3} (8 p^{3})$ and $\frac{2}{3}$ .

$- \log_{3} (8 p^{3}) - (\frac{2}{3} \log_{3} (8 p^{3}))$

Step 8.2

Simplify $\frac{2}{3} \log_{3} (8 p^{3})$ by moving $\frac{2}{3}$ inside the logarithm.

$- \log_{3} (8 p^{3}) - \log_{3} ({(8 p^{3})}^{\frac{2}{3}})$

Step 9

Apply the product rule to $8 p^{3}$ .

$- \log_{3} (8 p^{3}) - \log_{3} (8^{\frac{2}{3}} {(p^{3})}^{\frac{2}{3}})$

Step 10

Rewrite $8$ as $2^{3}$ .

$- \log_{3} (8 p^{3}) - \log_{3} ({(2^{3})}^{\frac{2}{3}} {(p^{3})}^{\frac{2}{3}})$

Step 11

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

$- \log_{3} (8 p^{3}) - \log_{3} (2^{3 (\frac{2}{3})} {(p^{3})}^{\frac{2}{3}})$

Step 12

Cancel the common factor of $3$ .

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Step 12.1

Cancel the common factor.

$- \log_{3} (8 p^{3}) - \log_{3} (2^{3 (\frac{2}{3})} {(p^{3})}^{\frac{2}{3}})$

Step 12.2

Rewrite the expression.

$- \log_{3} (8 p^{3}) - \log_{3} (2^{2} {(p^{3})}^{\frac{2}{3}})$

Step 13

Raise $2$ to the power of $2$ .

$- \log_{3} (8 p^{3}) - \log_{3} (4 {(p^{3})}^{\frac{2}{3}})$

Step 14

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

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Step 14.1

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

$- \log_{3} (8 p^{3}) - \log_{3} (4 p^{3 (\frac{2}{3})})$

Step 14.2

Cancel the common factor of $3$ .

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Step 14.2.1

Cancel the common factor.

$- \log_{3} (8 p^{3}) - \log_{3} (4 p^{3 (\frac{2}{3})})$

Step 14.2.2

Rewrite the expression.

$- \log_{3} (8 p^{3}) - \log_{3} (4 p^{2})$