Precalculus Examples

$\log ({(\frac{n^{5} \sqrt{m}}{k})}^{5})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m}$ as $m^{\frac{1}{2}}$ .

$\log ({(\frac{n^{5} m^{\frac{1}{2}}}{k})}^{5})$

Step 2

Expand $\log ({(\frac{n^{5} m^{\frac{1}{2}}}{k})}^{5})$ by moving $5$ outside the logarithm.

$5 \log (\frac{n^{5} m^{\frac{1}{2}}}{k})$

Step 3

Rewrite $\log (\frac{n^{5} m^{\frac{1}{2}}}{k})$ as $\log (n^{5} m^{\frac{1}{2}}) - \log (k)$ .

$5 (\log (n^{5} m^{\frac{1}{2}}) - \log (k))$

Step 4

Rewrite $\log (n^{5} m^{\frac{1}{2}})$ as $\log (n^{5}) + \log (m^{\frac{1}{2}})$ .

$5 (\log (n^{5}) + \log (m^{\frac{1}{2}}) - \log (k))$

Step 5

Expand $\log (n^{5})$ by moving $5$ outside the logarithm.

$5 (5 \log (n) + \log (m^{\frac{1}{2}}) - \log (k))$

Step 6

Expand $\log (m^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$5 (5 \log (n) + \frac{1}{2} \log (m) - \log (k))$

Step 7

Combine $\frac{1}{2}$ and $\log (m)$ .

$5 (5 \log (n) + \frac{\log (m)}{2} - \log (k))$

Step 8

Apply the distributive property.

$5 (5 \log (n)) + 5 \frac{\log (m)}{2} + 5 (- \log (k))$

Step 9

Simplify.

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Step 9.1

Multiply $5$ by $5$ .

$25 \log (n) + 5 \frac{\log (m)}{2} + 5 (- \log (k))$

Step 9.2

Combine $5$ and $\frac{\log (m)}{2}$ .

$25 \log (n) + \frac{5 \log (m)}{2} + 5 (- \log (k))$

Step 9.3

Multiply $- 1$ by $5$ .

$25 \log (n) + \frac{5 \log (m)}{2} - 5 \log (k)$