Precalculus Examples

$\log_{7} (\frac{\sqrt{x} \sqrt[7]{y}}{w^{3} \sqrt{z}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $14$ .

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Step 1.1.1

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

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

Step 1.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{7}{14}}$ .

$\log_{7} (\frac{x^{\frac{7}{14}} \sqrt[7]{y}}{w^{3} \sqrt{z}})$

Step 1.1.3

Rewrite $x^{\frac{7}{14}}$ as $\sqrt[14]{x^{7}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7}} \sqrt[7]{y}}{w^{3} \sqrt{z}})$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{y}$ as $y^{\frac{1}{7}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7}} y^{\frac{1}{7}}}{w^{3} \sqrt{z}})$

Step 1.1.5

Rewrite $y^{\frac{1}{7}}$ as $y^{\frac{2}{14}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7}} y^{\frac{2}{14}}}{w^{3} \sqrt{z}})$

Step 1.1.6

Rewrite $y^{\frac{2}{14}}$ as $\sqrt[14]{y^{2}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7}} \sqrt[14]{y^{2}}}{w^{3} \sqrt{z}})$

Step 1.2

Combine using the product rule for radicals.

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}}}{w^{3} \sqrt{z}})$

Step 2

Multiply $\frac{\sqrt[14]{x^{7} y^{2}}}{w^{3} \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}}}{w^{3} \sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[14]{x^{7} y^{2}}}{w^{3} \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} \sqrt{z} \sqrt{z}})$

Step 3.2

Move $\sqrt{z}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} (\sqrt{z} \sqrt{z})})$

Step 3.3

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} ({\sqrt{z}}^{1} \sqrt{z})})$

Step 3.4

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} ({\sqrt{z}}^{1} {\sqrt{z}}^{1})})$

Step 3.5

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} {\sqrt{z}}^{1 + 1}})$

Step 3.6

Add $1$ and $1$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} {\sqrt{z}}^{2}})$

Step 3.7

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

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Step 3.7.1

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} {(z^{\frac{1}{2}})}^{2}})$

Step 3.7.2

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} z^{\frac{1}{2} \cdot 2}})$

Step 3.7.3

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} z^{\frac{2}{2}}})$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} z^{\frac{2}{2}}})$

Step 3.7.4.2

Rewrite the expression.

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} z^{1}})$

Step 3.7.5

Simplify.

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt{z}}{w^{3} z})$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $14$ .

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Step 4.1.1

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

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} z^{\frac{1}{2}}}{w^{3} z})$

Step 4.1.2

Rewrite $z^{\frac{1}{2}}$ as $z^{\frac{7}{14}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} z^{\frac{7}{14}}}{w^{3} z})$

Step 4.1.3

Rewrite $z^{\frac{7}{14}}$ as $\sqrt[14]{z^{7}}$ .

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2}} \sqrt[14]{z^{7}}}{w^{3} z})$

Step 4.2

Combine using the product rule for radicals.

$\log_{7} (\frac{\sqrt[14]{x^{7} y^{2} z^{7}}}{w^{3} z})$

Step 5

Rewrite $\log_{7} (\frac{\sqrt[14]{x^{7} y^{2} z^{7}}}{w^{3} z})$ 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 $b = 10$ .

$\frac{\log (\frac{\sqrt[14]{x^{7} y^{2} z^{7}}}{w^{3} z})}{\log (7)}$