Algebra Examples

$\log_{b} (\frac{\sqrt{x}}{\sqrt[3]{z}})$

Step 1

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

$\log_{b} (\frac{\sqrt{x}}{\sqrt[3]{z}} \cdot \frac{{\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{2}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

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

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{\sqrt[3]{z} {\sqrt[3]{z}}^{2}})$

Step 2.2

Raise $\sqrt[3]{z}$ to the power of $1$ .

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

Step 2.3

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

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{1 + 2}})$

Step 2.4

Add $1$ and $2$ .

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{3}})$

Step 2.5

Rewrite ${\sqrt[3]{z}}^{3}$ as $z$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z}$ as $z^{\frac{1}{3}}$ .

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

Step 2.5.2

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

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{z^{\frac{1}{3} \cdot 3}})$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{z^{\frac{3}{3}}})$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{z^{\frac{3}{3}}})$

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

$\log_{b} (\frac{\sqrt{x} {\sqrt[3]{z}}^{2}}{z})$

Step 3

Rewrite ${\sqrt[3]{z}}^{2}$ as $\sqrt[3]{z^{2}}$ .

$\log_{b} (\frac{\sqrt{x} \sqrt[3]{z^{2}}}{z})$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

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

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

Step 4.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$\log_{b} (\frac{x^{\frac{3}{6}} \sqrt[3]{z^{2}}}{z})$

Step 4.1.3

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$\log_{b} (\frac{\sqrt[6]{x^{3}} \sqrt[3]{z^{2}}}{z})$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z^{2}}$ as $z^{\frac{2}{3}}$ .

$\log_{b} (\frac{\sqrt[6]{x^{3}} z^{\frac{2}{3}}}{z})$

Step 4.1.5

Rewrite $z^{\frac{2}{3}}$ as $z^{\frac{4}{6}}$ .

$\log_{b} (\frac{\sqrt[6]{x^{3}} z^{\frac{4}{6}}}{z})$

Step 4.1.6

Rewrite $z^{\frac{4}{6}}$ as $\sqrt[6]{z^{4}}$ .

$\log_{b} (\frac{\sqrt[6]{x^{3}} \sqrt[6]{z^{4}}}{z})$

Step 4.2

Combine using the product rule for radicals.

$\log_{b} (\frac{\sqrt[6]{x^{3} z^{4}}}{z})$

Step 5

Rewrite $\log_{b} (\frac{\sqrt[6]{x^{3} z^{4}}}{z})$ 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 $b = 10$ .

$\frac{\log (\frac{\sqrt[6]{x^{3} z^{4}}}{z})}{\log (b)}$