Precalculus Examples

$\log_{b} ({(\sqrt{\frac{x^{9}}{y^{7} z^{9}}})}^{\frac{1}{7}})$

Step 1

Rewrite $\frac{x^{9}}{y^{7} z^{9}}$ as ${(\frac{x^{4}}{y^{3} z^{4}})}^{2} \frac{x}{y z}$ .

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Step 1.1

Factor the perfect power ${(x^{4})}^{2}$ out of $x^{9}$ .

$\log_{b} ({\sqrt{\frac{{(x^{4})}^{2} x}{y^{7} z^{9}}}}^{\frac{1}{7}})$

Step 1.2

Factor the perfect power ${(y^{3} z^{4})}^{2}$ out of $y^{7} z^{9}$ .

$\log_{b} ({\sqrt{\frac{{(x^{4})}^{2} x}{{(y^{3} z^{4})}^{2} (y z)}}}^{\frac{1}{7}})$

Step 1.3

Rearrange the fraction $\frac{{(x^{4})}^{2} x}{{(y^{3} z^{4})}^{2} (y z)}$ .

$\log_{b} ({\sqrt{{(\frac{x^{4}}{y^{3} z^{4}})}^{2} \frac{x}{y z}}}^{\frac{1}{7}})$

Step 2

Pull terms out from under the radical.

$\log_{b} ({(\frac{x^{4}}{y^{3} z^{4}} \sqrt{\frac{x}{y z}})}^{\frac{1}{7}})$

Step 3

Rewrite $\sqrt{\frac{x}{y z}}$ as $\frac{\sqrt{x}}{\sqrt{y z}}$ .

$\log_{b} ({(\frac{x^{4}}{y^{3} z^{4}} \cdot \frac{\sqrt{x}}{\sqrt{y z}})}^{\frac{1}{7}})$

Step 4

Combine.

$\log_{b} ({(\frac{x^{4} \sqrt{x}}{y^{3} z^{4} \sqrt{y z}})}^{\frac{1}{7}})$

Step 5

Multiply $\frac{x^{4} \sqrt{x}}{y^{3} z^{4} \sqrt{y z}}$ by $\frac{\sqrt{y z}}{\sqrt{y z}}$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x}}{y^{3} z^{4} \sqrt{y z}} \cdot \frac{\sqrt{y z}}{\sqrt{y z}})}^{\frac{1}{7}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{x^{4} \sqrt{x}}{y^{3} z^{4} \sqrt{y z}}$ by $\frac{\sqrt{y z}}{\sqrt{y z}}$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} \sqrt{y z} \sqrt{y z}})}^{\frac{1}{7}})$

Step 6.2

Move $\sqrt{y z}$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} (\sqrt{y z} \sqrt{y z})})}^{\frac{1}{7}})$

Step 6.3

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} ({\sqrt{y z}}^{1} \sqrt{y z})})}^{\frac{1}{7}})$

Step 6.4

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} ({\sqrt{y z}}^{1} {\sqrt{y z}}^{1})})}^{\frac{1}{7}})$

Step 6.5

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {\sqrt{y z}}^{1 + 1}})}^{\frac{1}{7}})$

Step 6.6

Add $1$ and $1$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {\sqrt{y z}}^{2}})}^{\frac{1}{7}})$

Step 6.7

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

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Step 6.7.1

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {({(y z)}^{\frac{1}{2}})}^{2}})}^{\frac{1}{7}})$

Step 6.7.2

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {(y z)}^{\frac{1}{2} \cdot 2}})}^{\frac{1}{7}})$

Step 6.7.3

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {(y z)}^{\frac{2}{2}}})}^{\frac{1}{7}})$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {(y z)}^{\frac{2}{2}}})}^{\frac{1}{7}})$

Step 6.7.4.2

Rewrite the expression.

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} {(y z)}^{1}})}^{\frac{1}{7}})$

Step 6.7.5

Simplify.

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{3} z^{4} (y z)})}^{\frac{1}{7}})$

Step 7

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 7.1

Move $y$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y \cdot y^{3} z^{4} z})}^{\frac{1}{7}})$

Step 7.2

Multiply $y$ by $y^{3}$ .

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Step 7.2.1

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{1} y^{3} z^{4} z})}^{\frac{1}{7}})$

Step 7.2.2

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{1 + 3} z^{4} z})}^{\frac{1}{7}})$

Step 7.3

Add $1$ and $3$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{4} z^{4} z})}^{\frac{1}{7}})$

Step 8

Multiply $z^{4}$ by $z$ by adding the exponents.

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Step 8.1

Move $z$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{4} (z \cdot z^{4})})}^{\frac{1}{7}})$

Step 8.2

Multiply $z$ by $z^{4}$ .

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Step 8.2.1

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{4} (z^{1} z^{4})})}^{\frac{1}{7}})$

Step 8.2.2

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

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{4} z^{1 + 4}})}^{\frac{1}{7}})$

Step 8.3

Add $1$ and $4$ .

$\log_{b} ({(\frac{x^{4} \sqrt{x} \sqrt{y z}}{y^{4} z^{5}})}^{\frac{1}{7}})$

Step 9

Combine using the product rule for radicals.

$\log_{b} ({(\frac{x^{4} \sqrt{y z x}}{y^{4} z^{5}})}^{\frac{1}{7}})$

Step 10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 10.1

Apply the product rule to $\frac{x^{4} \sqrt{y z x}}{y^{4} z^{5}}$ .

$\log_{b} (\frac{{(x^{4} \sqrt{y z x})}^{\frac{1}{7}}}{{(y^{4} z^{5})}^{\frac{1}{7}}})$

Step 10.2

Apply the product rule to $x^{4} \sqrt{y z x}$ .

$\log_{b} (\frac{{(x^{4})}^{\frac{1}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{{(y^{4} z^{5})}^{\frac{1}{7}}})$

Step 10.3

Apply the product rule to $y^{4} z^{5}$ .

$\log_{b} (\frac{{(x^{4})}^{\frac{1}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{{(y^{4})}^{\frac{1}{7}} {(z^{5})}^{\frac{1}{7}}})$

Step 11

Multiply the exponents in ${(x^{4})}^{\frac{1}{7}}$ .

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Step 11.1

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

$\log_{b} (\frac{x^{4 (\frac{1}{7})} {\sqrt{y z x}}^{\frac{1}{7}}}{{(y^{4})}^{\frac{1}{7}} {(z^{5})}^{\frac{1}{7}}})$

Step 11.2

Combine $4$ and $\frac{1}{7}$ .

$\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{{(y^{4})}^{\frac{1}{7}} {(z^{5})}^{\frac{1}{7}}})$

Step 12

Simplify the denominator.

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Step 12.1

Multiply the exponents in ${(y^{4})}^{\frac{1}{7}}$ .

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Step 12.1.1

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

$\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{4 (\frac{1}{7})} {(z^{5})}^{\frac{1}{7}}})$

Step 12.1.2

Combine $4$ and $\frac{1}{7}$ .

$\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{\frac{4}{7}} {(z^{5})}^{\frac{1}{7}}})$

Step 12.2

Multiply the exponents in ${(z^{5})}^{\frac{1}{7}}$ .

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Step 12.2.1

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

$\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{\frac{4}{7}} z^{5 (\frac{1}{7})}})$

Step 12.2.2

Combine $5$ and $\frac{1}{7}$ .

$\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{\frac{4}{7}} z^{\frac{5}{7}}})$

Step 13

Rewrite $\log_{b} (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{\frac{4}{7}} z^{\frac{5}{7}}})$ using the change of base formula.

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Step 13.1

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 13.2

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

$\frac{\log (\frac{x^{\frac{4}{7}} {\sqrt{y z x}}^{\frac{1}{7}}}{y^{\frac{4}{7}} z^{\frac{5}{7}}})}{\log (b)}$