Precalculus Examples

$\frac{\log_{3} (x^{5} y^{7})}{\sqrt{z}}$

Step 1

Multiply $\frac{\log_{3} (x^{5} y^{7})}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\frac{\log_{3} (x^{5} y^{7})}{\sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\log_{3} (x^{5} y^{7})}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{\sqrt{z} \sqrt{z}}$

Step 2.2

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{{\sqrt{z}}^{1} \sqrt{z}}$

Step 2.3

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{{\sqrt{z}}^{1} {\sqrt{z}}^{1}}$

Step 2.4

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{{\sqrt{z}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{{\sqrt{z}}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{{(z^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{z^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{z^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{z^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{z^{1}}$

Step 2.6.5

Simplify.

$\frac{\log_{3} (x^{5} y^{7}) \sqrt{z}}{z}$

Step 3

Rewrite $\log_{3} (x^{5} y^{7})$ using the change of base formula.

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Step 3.1

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 $0$ .

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

Step 3.2

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

$\frac{\frac{\log (x^{5} y^{7})}{\log (3)} \sqrt{z}}{z}$

Step 4

Combine $\frac{\log (x^{5} y^{7})}{\log (3)}$ and $\sqrt{z}$ .

$\frac{\frac{\log (x^{5} y^{7}) \sqrt{z}}{\log (3)}}{z}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (x^{5} y^{7}) \sqrt{z}}{\log (3)} \cdot \frac{1}{z}$

Step 6

Combine.

$\frac{\log (x^{5} y^{7}) \sqrt{z} \cdot 1}{\log (3) z}$

Step 7

Simplify the expression.

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Step 7.1

Multiply $\log (x^{5} y^{7})$ by $1$ .

$\frac{\log (x^{5} y^{7}) \sqrt{z}}{\log (3) z}$

Step 7.2

Reorder factors in $\frac{\log (x^{5} y^{7}) \sqrt{z}}{\log (3) z}$ .

$\frac{\log (x^{5} y^{7}) \sqrt{z}}{z \log (3)}$