Expand the Logarithmic Expression log base 3 of 1/( square root of 3)

$\log_{3} (\frac{1}{\sqrt{3}})$

Step 1

Rewrite $\log_{3} (\frac{1}{\sqrt{3}})$ as $\log_{3} (1) - \log_{3} (\sqrt{3})$ .

$\log_{3} (1) - \log_{3} (\sqrt{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3

Expand $\log_{3} (3^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

Logarithm base $3$ of $1$ is $0$ .

$0 - (\frac{1}{2} \log_{3} (3))$

Step 5

Logarithm base $3$ of $3$ is $1$ .

$0 - (\frac{1}{2} \cdot 1)$

Step 6

Multiply $\frac{1}{2}$ by $1$ .

$0 - \frac{1}{2}$

Step 7

Subtract $\frac{1}{2}$ from $0$ .

$- \frac{1}{2}$