Precalculus Examples

$\log_{b} ({(4 m n)}^{5})$

Step 1

Expand $\log_{b} ({(4 m n)}^{5})$ by moving $5$ outside the logarithm.

$5 \log_{b} (4 m n)$

Step 2

Rewrite $\log_{b} (4 m n)$ as $\log_{b} (4 m) + \log_{b} (n)$ .

$5 (\log_{b} (4 m) + \log_{b} (n))$

Step 3

Rewrite $\log_{b} (4 m)$ as $\log_{b} (4) + \log_{b} (m)$ .

$5 (\log_{b} (4) + \log_{b} (m) + \log_{b} (n))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{b} (4)$ as $\log_{b} (2^{2})$ .

$5 (\log_{b} (2^{2}) + \log_{b} (m) + \log_{b} (n))$

Step 4.2

Expand $\log_{b} (2^{2})$ by moving $2$ outside the logarithm.

$5 (2 \log_{b} (2) + \log_{b} (m) + \log_{b} (n))$

Step 5

Apply the distributive property.

$5 (2 \log_{b} (2)) + 5 \log_{b} (m) + 5 \log_{b} (n)$

Step 6

Multiply $2$ by $5$ .

$10 \log_{b} (2) + 5 \log_{b} (m) + 5 \log_{b} (n)$