Evaluate log base 17 of 1/( square root of 17)

$\log_{17} (\frac{1}{\sqrt{17}})$

Step 1

Multiply $\frac{1}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\log_{17} (\frac{1}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}})$

Step 2

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\log_{17} (\frac{\sqrt{17}}{\sqrt{17} \sqrt{17}})$

Raise $\sqrt{17}$ to the power of $1$ .

$\log_{17} (\frac{\sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}})$

Raise $\sqrt{17}$ to the power of $1$ .

$\log_{17} (\frac{\sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}})$

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

$\log_{17} (\frac{\sqrt{17}}{{\sqrt{17}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{17} (\frac{\sqrt{17}}{{\sqrt{17}}^{2}})$

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\log_{17} (\frac{\sqrt{17}}{{(17^{\frac{1}{2}})}^{2}})$

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

$\log_{17} (\frac{\sqrt{17}}{17^{\frac{1}{2} \cdot 2}})$

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

$\log_{17} (\frac{\sqrt{17}}{17^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{17} (\frac{\sqrt{17}}{17^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{17} (\frac{\sqrt{17}}{17^{1}})$

Evaluate the exponent.

$\log_{17} (\frac{\sqrt{17}}{17})$

Step 3

Rewrite $\frac{\sqrt{17}}{17}$ as $\sqrt{17} \cdot 17^{- 1}$ .

$\log_{17} (\sqrt{17} \cdot 17^{- 1})$

Step 4

Rewrite $\log_{17} (\sqrt{17} \cdot 17^{- 1})$ as $\log_{17} (\sqrt{17}) + \log_{17} (17^{- 1})$ .

$\log_{17} (\sqrt{17}) + \log_{17} (17^{- 1})$

Step 5

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

$\log_{17} (\sqrt{17}) - \log_{17} (17)$

Step 6

Logarithm base $17$ of $17$ is $1$ .

$\log_{17} (\sqrt{17}) - 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$\log_{17} (\sqrt{17}) - 1$

Step 8

Rewrite $\log_{17} (\sqrt{17})$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\sqrt{17})}{\log (17)} - 1$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{17})}{\log (17)} - 1$

Decimal Form:

$- 0.5$

Evaluate log base 17 of 1/( square root of 17)

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