Precalculus Examples

$\frac{\log_{6} (\sqrt{x} y^{3})}{2^{3}}$

Step 1

Raise $2$ to the power of $3$ .

$\frac{\log_{6} (\sqrt{x} y^{3})}{8}$

Step 2

Rewrite $\frac{\log_{6} (\sqrt{x} y^{3})}{8}$ as $\frac{1}{8} \log_{6} (\sqrt{x} y^{3})$ .

$\frac{1}{8} \log_{6} (\sqrt{x} y^{3})$

Step 3

Simplify $\frac{1}{8} \log_{6} (\sqrt{x} y^{3})$ by moving $\frac{1}{8}$ inside the logarithm.

$\log_{6} ({(\sqrt{x} y^{3})}^{\frac{1}{8}})$

Step 4

Apply the product rule to $\sqrt{x} y^{3}$ .

$\log_{6} ({\sqrt{x}}^{\frac{1}{8}} {(y^{3})}^{\frac{1}{8}})$

Step 5

Multiply the exponents in ${(y^{3})}^{\frac{1}{8}}$ .

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Step 5.1

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

$\log_{6} ({\sqrt{x}}^{\frac{1}{8}} y^{3 (\frac{1}{8})})$

Step 5.2

Combine $3$ and $\frac{1}{8}$ .

$\log_{6} ({\sqrt{x}}^{\frac{1}{8}} y^{\frac{3}{8}})$

Step 6

Rewrite $\log_{6} ({\sqrt{x}}^{\frac{1}{8}} y^{\frac{3}{8}})$ using the change of base formula.

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Step 6.1

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 6.2

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

$\frac{\log ({\sqrt{x}}^{\frac{1}{8}} y^{\frac{3}{8}})}{\log (6)}$