Precalculus Examples

$\log_{b} ({(\frac{m z^{3}}{n})}^{\frac{3}{5}})$

Step 1

Expand $\log_{b} ({(\frac{m z^{3}}{n})}^{\frac{3}{5}})$ by moving $\frac{3}{5}$ outside the logarithm.

$\frac{3}{5} \log_{b} (\frac{m z^{3}}{n})$

Step 2

Rewrite $\log_{b} (\frac{m z^{3}}{n})$ as $\log_{b} (m z^{3}) - \log_{b} (n)$ .

$\frac{3}{5} (\log_{b} (m z^{3}) - \log_{b} (n))$

Step 3

Rewrite $\log_{b} (m z^{3})$ as $\log_{b} (m) + \log_{b} (z^{3})$ .

$\frac{3}{5} (\log_{b} (m) + \log_{b} (z^{3}) - \log_{b} (n))$

Step 4

Expand $\log_{b} (z^{3})$ by moving $3$ outside the logarithm.

$\frac{3}{5} (\log_{b} (m) + 3 \log_{b} (z) - \log_{b} (n))$

Step 5

Apply the distributive property.

$\frac{3}{5} \log_{b} (m) + \frac{3}{5} (3 \log_{b} (z)) + \frac{3}{5} (- \log_{b} (n))$

Step 6

Simplify.

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Step 6.1

Combine $\frac{3}{5}$ and $\log_{b} (m)$ .

$\frac{3 \log_{b} (m)}{5} + \frac{3}{5} (3 \log_{b} (z)) + \frac{3}{5} (- \log_{b} (n))$

Step 6.2

Multiply $\frac{3}{5} (3 \log_{b} (z))$ .

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Step 6.2.1

Combine $3$ and $\frac{3}{5}$ .

$\frac{3 \log_{b} (m)}{5} + \frac{3 \cdot 3}{5} \log_{b} (z) + \frac{3}{5} (- \log_{b} (n))$

Step 6.2.2

Multiply $3$ by $3$ .

$\frac{3 \log_{b} (m)}{5} + \frac{9}{5} \log_{b} (z) + \frac{3}{5} (- \log_{b} (n))$

Step 6.2.3

Combine $\frac{9}{5}$ and $\log_{b} (z)$ .

$\frac{3 \log_{b} (m)}{5} + \frac{9 \log_{b} (z)}{5} + \frac{3}{5} (- \log_{b} (n))$

Step 6.3

Combine $\frac{3}{5}$ and $\log_{b} (n)$ .

$\frac{3 \log_{b} (m)}{5} + \frac{9 \log_{b} (z)}{5} - \frac{3 \log_{b} (n)}{5}$