Simplify/Condense log base 7 of 1/( square root of 7)

$\log_{7} (\frac{1}{\sqrt{7}})$

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\log_{7} (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\log_{7} (\frac{\sqrt{7}}{\sqrt{7} \sqrt{7}})$

Raise $\sqrt{7}$ to the power of $1$ .

$\log_{7} (\frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}})$

Raise $\sqrt{7}$ to the power of $1$ .

$\log_{7} (\frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{7} (\frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{7} (\frac{\sqrt{7}}{{\sqrt{7}}^{2}})$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\log_{7} (\frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{7} (\frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

$\log_{7} (\frac{\sqrt{7}}{7^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{7} (\frac{\sqrt{7}}{7^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{7} (\frac{\sqrt{7}}{7^{1}})$

Evaluate the exponent.

$\log_{7} (\frac{\sqrt{7}}{7})$

Rewrite $\frac{\sqrt{7}}{7}$ as $\sqrt{7} \cdot 7^{- 1}$ .

$\log_{7} (\sqrt{7} \cdot 7^{- 1})$

Rewrite $\log_{7} (\sqrt{7} \cdot 7^{- 1})$ as $\log_{7} (\sqrt{7}) + \log_{7} (7^{- 1})$ .

$\log_{7} (\sqrt{7}) + \log_{7} (7^{- 1})$

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{7} (\sqrt{7}) - \log_{7} (7)$

Logarithm base $7$ of $7$ is $1$ .

$\log_{7} (\sqrt{7}) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$\log_{7} (\sqrt{7}) - 1$

The result can be shown in multiple forms.

Exact Form:

$\log_{7} (\sqrt{7}) - 1$

Decimal Form:

$- 0.5$