Precalculus Examples

$4 \log_{a} (6 x^{9}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log_{a} (6 x^{9})$ by moving $4$ inside the logarithm.

$\log_{a} ({(6 x^{9})}^{4}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1.2

Apply the product rule to $6 x^{9}$ .

$\log_{a} (6^{4} {(x^{9})}^{4}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1.3

Raise $6$ to the power of $4$ .

$\log_{a} (1296 {(x^{9})}^{4}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1.4

Multiply the exponents in ${(x^{9})}^{4}$ .

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Step 1.4.1

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

$\log_{a} (1296 x^{9 \cdot 4}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1.4.2

Multiply $9$ by $4$ .

$\log_{a} (1296 x^{36}) - \frac{1}{7} \cdot \log_{a} (8 x + 11)$

Step 1.5

Multiply $- \frac{1}{7} \log_{a} (8 x + 11)$ .

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Step 1.5.1

Reorder $\log_{a} (8 x + 11)$ and $\frac{1}{7}$ .

$\log_{a} (1296 x^{36}) - (\frac{1}{7} \log_{a} (8 x + 11))$

Step 1.5.2

Simplify $\frac{1}{7} \log_{a} (8 x + 11)$ by moving $\frac{1}{7}$ inside the logarithm.

$\log_{a} (1296 x^{36}) - \log_{a} ({(8 x + 11)}^{\frac{1}{7}})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{a} (\frac{1296 x^{36}}{{(8 x + 11)}^{\frac{1}{7}}})$