Precalculus Examples

$\log (\sqrt{x^{3} - 9}) = 2$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} - 9}$ as ${(x^{3} - 9)}^{\frac{1}{2}}$ .

$\log ({(x^{3} - 9)}^{\frac{1}{2}}) = 2$

Step 2

Rewrite $\log ({(x^{3} - 9)}^{\frac{1}{2}}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{2} = {(x^{3} - 9)}^{\frac{1}{2}}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as ${(x^{3} - 9)}^{\frac{1}{2}} = 10^{2}$ .

${(x^{3} - 9)}^{\frac{1}{2}} = 10^{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x^{3} - 9)}^{\frac{1}{2}})}^{2} = {(10^{2})}^{2}$

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${({(x^{3} - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

Multiply the exponents in ${({(x^{3} - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

${(x^{3} - 9)}^{\frac{1}{2} \cdot 2} = {(10^{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

${(x^{3} - 9)}^{\frac{1}{2} \cdot 2} = {(10^{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

${(x^{3} - 9)}^{1} = {(10^{2})}^{2}$

Step 3.3.1.1.2

Simplify.

$x^{3} - 9 = {(10^{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(10^{2})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${(10^{2})}^{2}$ .

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Step 3.3.2.1.1.1

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

$x^{3} - 9 = 10^{2 \cdot 2}$

Step 3.3.2.1.1.2

Multiply $2$ by $2$ .

$x^{3} - 9 = 10^{4}$

Step 3.3.2.1.2

Raise $10$ to the power of $4$ .

$x^{3} - 9 = 10000$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.4.1.1

Add $9$ to both sides of the equation.

$x^{3} = 10000 + 9$

Step 3.4.1.2

Add $10000$ and $9$ .

$x^{3} = 10009$

Step 3.4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{10009}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[3]{10009}$

Decimal Form:

$x = 21.55080826 \dots$