Precalculus Examples

$\frac{\log_{z} (x y^{z})}{64}$

Step 1

Rewrite $\frac{\log_{z} (x y^{z})}{64}$ as $\frac{1}{64} \log_{z} (x y^{z})$ .

$\frac{1}{64} \log_{z} (x y^{z})$

Step 2

Simplify $\frac{1}{64} \log_{z} (x y^{z})$ by moving $\frac{1}{64}$ inside the logarithm.

$\log_{z} ({(x y^{z})}^{\frac{1}{64}})$

Step 3

Apply the product rule to $x y^{z}$ .

$\log_{z} (x^{\frac{1}{64}} {(y^{z})}^{\frac{1}{64}})$

Step 4

Multiply the exponents in ${(y^{z})}^{\frac{1}{64}}$ .

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Step 4.1

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

$\log_{z} (x^{\frac{1}{64}} y^{z \frac{1}{64}})$

Step 4.2

Combine $z$ and $\frac{1}{64}$ .

$\log_{z} (x^{\frac{1}{64}} y^{\frac{z}{64}})$

Step 5

Rewrite $\log_{z} (x^{\frac{1}{64}} y^{\frac{z}{64}})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (x^{\frac{1}{64}} y^{\frac{z}{64}})}{\log (z)}$