Expand the Logarithmic Expression log base a of square root of 80

$\log_{a} (\sqrt{80})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{80}$ as $80^{\frac{1}{2}}$ .

$\log_{a} (80^{\frac{1}{2}})$

Step 2

Expand $\log_{a} (80^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{a} (80)$

Step 3

Rewrite $\log_{a} (80)$ as $\log_{a} (2^{4} \cdot 5)$ .

$\frac{1}{2} \log_{a} (2^{4} \cdot 5)$

Step 4

Rewrite $\log_{a} (2^{4} \cdot 5)$ as $\log_{a} (2^{4}) + \log_{a} (5)$ .

$\frac{1}{2} (\log_{a} (2^{4}) + \log_{a} (5))$

Step 5

Expand $\log_{a} (2^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} (4 \log_{a} (2) + \log_{a} (5))$

Step 6

Apply the distributive property.

$\frac{1}{2} (4 \log_{a} (2)) + \frac{1}{2} \log_{a} (5)$

Step 7

Cancel the common factor of $2$ .

Factor $2$ out of $4 \log_{a} (2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

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

$2 \log_{a} (2) + \frac{\log_{a} (5)}{2}$