Precalculus Examples

$\log_{4} (\frac{\sqrt[9]{m} \sqrt[5]{n}}{k^{2}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $45$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{m}$ as $m^{\frac{1}{9}}$ .

$\log_{4} (\frac{m^{\frac{1}{9}} \sqrt[5]{n}}{k^{2}})$

Step 1.1.2

Rewrite $m^{\frac{1}{9}}$ as $m^{\frac{5}{45}}$ .

$\log_{4} (\frac{m^{\frac{5}{45}} \sqrt[5]{n}}{k^{2}})$

Step 1.1.3

Rewrite $m^{\frac{5}{45}}$ as $\sqrt[45]{m^{5}}$ .

$\log_{4} (\frac{\sqrt[45]{m^{5}} \sqrt[5]{n}}{k^{2}})$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{n}$ as $n^{\frac{1}{5}}$ .

$\log_{4} (\frac{\sqrt[45]{m^{5}} n^{\frac{1}{5}}}{k^{2}})$

Step 1.1.5

Rewrite $n^{\frac{1}{5}}$ as $n^{\frac{9}{45}}$ .

$\log_{4} (\frac{\sqrt[45]{m^{5}} n^{\frac{9}{45}}}{k^{2}})$

Step 1.1.6

Rewrite $n^{\frac{9}{45}}$ as $\sqrt[45]{n^{9}}$ .

$\log_{4} (\frac{\sqrt[45]{m^{5}} \sqrt[45]{n^{9}}}{k^{2}})$

Step 1.2

Combine using the product rule for radicals.

$\log_{4} (\frac{\sqrt[45]{m^{5} n^{9}}}{k^{2}})$

Step 2

Rewrite $\log_{4} (\frac{\sqrt[45]{m^{5} n^{9}}}{k^{2}})$ using the change of base formula.

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Step 2.1

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 $0$ .

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

Step 2.2

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

$\frac{\log (\frac{\sqrt[45]{m^{5} n^{9}}}{k^{2}})}{\log (4)}$