Precalculus Examples

$4 \log_{a} (5 x^{9}) - \frac{1}{7} \cdot \log_{a} (2 x + 5)$

Step 1

Rewrite $\log_{a} (5 x^{9})$ as $\log_{a} (5) + \log_{a} (x^{9})$ .

$4 (\log_{a} (5) + \log_{a} (x^{9})) - \frac{1}{7} \cdot \log_{a} (2 x + 5)$

Step 2

Expand $\log_{a} (x^{9})$ by moving $9$ outside the logarithm.

$4 (\log_{a} (5) + 9 \log_{a} (x)) - \frac{1}{7} \cdot \log_{a} (2 x + 5)$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$4 \log_{a} (5) + 4 (9 \log_{a} (x)) - \frac{1}{7} \cdot \log_{a} (2 x + 5)$

Step 3.2

Multiply $9$ by $4$ .

$4 \log_{a} (5) + 36 \log_{a} (x) - \frac{1}{7} \cdot \log_{a} (2 x + 5)$

Step 3.3

Combine $\log_{a} (2 x + 5)$ and $\frac{1}{7}$ .

$4 \log_{a} (5) + 36 \log_{a} (x) - \frac{\log_{a} (2 x + 5)}{7}$

Step 4

To write $4 \log_{a} (5)$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$36 \log_{a} (x) + 4 \log_{a} (5) \cdot \frac{7}{7} - \frac{\log_{a} (2 x + 5)}{7}$

Step 5

Combine $4 \log_{a} (5)$ and $\frac{7}{7}$ .

$36 \log_{a} (x) + \frac{4 \log_{a} (5) \cdot 7}{7} - \frac{\log_{a} (2 x + 5)}{7}$

Step 6

Combine the numerators over the common denominator.

$36 \log_{a} (x) + \frac{4 \log_{a} (5) \cdot 7 - \log_{a} (2 x + 5)}{7}$

Step 7

Multiply $7$ by $4$ .

$36 \log_{a} (x) + \frac{28 \log_{a} (5) - \log_{a} (2 x + 5)}{7}$