Precalculus Examples

$\log_{5} (\frac{5 \sqrt{y}}{x^{2}})$

Step 1

Rewrite $\log_{5} (\frac{5 \sqrt{y}}{x^{2}})$ as $\log_{5} (5 \sqrt{y}) - \log_{5} (x^{2})$ .

$\log_{5} (5 \sqrt{y}) - \log_{5} (x^{2})$

Step 2

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

$\log_{5} (5 y^{\frac{1}{2}}) - \log_{5} (x^{2})$

Step 3

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

$\log_{5} (5 y^{\frac{1}{2}}) - (2 \log_{5} (x))$

Step 4

Multiply $2$ by $- 1$ .

$\log_{5} (5 y^{\frac{1}{2}}) - 2 \log_{5} (x)$

Step 5

Rewrite $\log_{5} (5 y^{\frac{1}{2}})$ as $\log_{5} (5) + \log_{5} (y^{\frac{1}{2}})$ .

$\log_{5} (5) + \log_{5} (y^{\frac{1}{2}}) - 2 \log_{5} (x)$

Step 6

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

$\log_{5} (5) + \frac{1}{2} \log_{5} (y) - 2 \log_{5} (x)$

Step 7

Logarithm base $5$ of $5$ is $1$ .

$1 + \frac{1}{2} \log_{5} (y) - 2 \log_{5} (x)$

Step 8

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

$1 + \frac{\log_{5} (y)}{2} - 2 \log_{5} (x)$