Expand the Logarithmic Expression log base 5 of ( square root of x)/125

$\log_{5} (\frac{\sqrt{x}}{125})$

Step 1

Rewrite $\log_{5} (\frac{\sqrt{x}}{125})$ as $\log_{5} (\sqrt{x}) - \log_{5} (125)$ .

$\log_{5} (\sqrt{x}) - \log_{5} (125)$

Step 2

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

$\log_{5} (x^{\frac{1}{2}}) - \log_{5} (125)$

Step 3

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

$\frac{1}{2} \log_{5} (x) - \log_{5} (125)$

Step 4

Logarithm base $5$ of $125$ is $3$ .

$\frac{1}{2} \log_{5} (x) - 1 \cdot 3$

Step 5

Multiply $- 1$ by $3$ .

$\frac{1}{2} \log_{5} (x) - 3$

Step 6

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

$\frac{\log_{5} (x)}{2} - 3$