Evaluate log base b of square root of 57/74

$\log_{b} (\sqrt{\frac{57}{74}})$

Step 1

Rewrite $\sqrt{\frac{57}{74}}$ as $\frac{\sqrt{57}}{\sqrt{74}}$ .

$\log_{b} (\frac{\sqrt{57}}{\sqrt{74}})$

Step 2

Multiply $\frac{\sqrt{57}}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\log_{b} (\frac{\sqrt{57}}{\sqrt{74}} \cdot \frac{\sqrt{74}}{\sqrt{74}})$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt{57}}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{\sqrt{74} \sqrt{74}})$

Raise $\sqrt{74}$ to the power of $1$ .

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1} \sqrt{74}})$

Raise $\sqrt{74}$ to the power of $1$ .

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1} {\sqrt{74}}^{1}})$

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

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{{\sqrt{74}}^{2}})$

Rewrite ${\sqrt{74}}^{2}$ as $74$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{74}$ as $74^{\frac{1}{2}}$ .

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{{(74^{\frac{1}{2}})}^{2}})$

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

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{74^{\frac{1}{2} \cdot 2}})$

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

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{74^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{74^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{74^{1}})$

Evaluate the exponent.

$\log_{b} (\frac{\sqrt{57} \sqrt{74}}{74})$

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

$\log_{b} (\frac{\sqrt{57 \cdot 74}}{74})$

Multiply $57$ by $74$ .

$\log_{b} (\frac{\sqrt{4218}}{74})$

Step 5

Rewrite $\log_{b} (\frac{\sqrt{4218}}{74})$ 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 $0$ .

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

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

$\frac{\log (\frac{\sqrt{4218}}{74})}{\log (b)}$