Precalculus Examples

$\log_{5} (M^{2} \sqrt{N})$

Step 1

Rewrite $\log_{5} (M^{2} \sqrt{N})$ as $\log_{5} (M^{2}) + \log_{5} (\sqrt{N})$ .

$\log_{5} (M^{2}) + \log_{5} (\sqrt{N})$

Step 2

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

$\log_{5} (M^{2}) + \log_{5} (N^{\frac{1}{2}})$

Step 3

Expand $\log_{5} (M^{2})$ by moving $2$ outside the logarithm.

$2 \log_{5} (M) + \log_{5} (N^{\frac{1}{2}})$

Step 4

Expand $\log_{5} (N^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$2 \log_{5} (M) + \frac{1}{2} \log_{5} (N)$

Step 5

Combine $\frac{1}{2}$ and $\log_{5} (N)$ .

$2 \log_{5} (M) + \frac{\log_{5} (N)}{2}$