Precalculus Examples

$\log (6) (\frac{x^{3}}{\sqrt{y z^{5}}})$

Step 1

Rewrite $\log (6)$ as $\log (2 (3))$ .

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

Step 2

Rewrite $\log (2 (3))$ as $\log (2) + \log (3)$ .

$(\log (2) + \log (3)) \frac{x^{3}}{\sqrt{y z^{5}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $y z^{5}$ as ${(z^{2})}^{2} (y z)$ .

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Step 3.1.1

Factor out $z^{4}$ .

$(\log (2) + \log (3)) \frac{x^{3}}{\sqrt{y (z^{4} z)}}$

Step 3.1.2

Rewrite $z^{4}$ as ${(z^{2})}^{2}$ .

$(\log (2) + \log (3)) \frac{x^{3}}{\sqrt{y ({(z^{2})}^{2} z)}}$

Step 3.1.3

Reorder $y$ and ${(z^{2})}^{2}$ .

$(\log (2) + \log (3)) \frac{x^{3}}{\sqrt{{(z^{2})}^{2} y z}}$

Step 3.1.4

Add parentheses.

$(\log (2) + \log (3)) \frac{x^{3}}{\sqrt{{(z^{2})}^{2} (y z)}}$

Step 3.2

Pull terms out from under the radical.

$(\log (2) + \log (3)) \frac{x^{3}}{z^{2} \sqrt{y z}}$

Step 4

Multiply $\frac{x^{3}}{z^{2} \sqrt{y z}}$ by $\frac{\sqrt{y z}}{\sqrt{y z}}$ .

$(\log (2) + \log (3)) (\frac{x^{3}}{z^{2} \sqrt{y z}} \cdot \frac{\sqrt{y z}}{\sqrt{y z}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{x^{3}}{z^{2} \sqrt{y z}}$ by $\frac{\sqrt{y z}}{\sqrt{y z}}$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} \sqrt{y z} \sqrt{y z}}$

Step 5.2

Move $\sqrt{y z}$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} (\sqrt{y z} \sqrt{y z})}$

Step 5.3

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} ({\sqrt{y z}}^{1} \sqrt{y z})}$

Step 5.4

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} ({\sqrt{y z}}^{1} {\sqrt{y z}}^{1})}$

Step 5.5

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {\sqrt{y z}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {\sqrt{y z}}^{2}}$

Step 5.7

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

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Step 5.7.1

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {({(y z)}^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {(y z)}^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {(y z)}^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {(y z)}^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} {(y z)}^{1}}$

Step 5.7.5

Simplify.

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{2} (y z)}$

Step 6

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 6.1

Move $z$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z \cdot z^{2} y}$

Step 6.2

Multiply $z$ by $z^{2}$ .

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Step 6.2.1

Raise $z$ to the power of $1$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{1} z^{2} y}$

Step 6.2.2

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

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{1 + 2} y}$

Step 6.3

Add $1$ and $2$ .

$(\log (2) + \log (3)) \frac{x^{3} \sqrt{y z}}{z^{3} y}$

Step 7

Multiply $\log (2) + \log (3)$ by $\frac{x^{3} \sqrt{y z}}{z^{3} y}$ .

$\frac{(\log (2) + \log (3)) (x^{3} \sqrt{y z})}{z^{3} y}$

Step 8

Reorder factors in $\frac{(\log (2) + \log (3)) x^{3} \sqrt{y z}}{z^{3} y}$ .

$\frac{x^{3} \sqrt{y z} (\log (2) + \log (3))}{z^{3} y}$