Precalculus Examples

$\log_{64} (x - 11) = \frac{2}{3}$

Step 1

Rewrite $\log_{64} (x - 11) = \frac{2}{3}$ 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 $b^{y} = x$ .

$64^{\frac{2}{3}} = x - 11$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x - 11 = 64^{\frac{2}{3}}$ .

$x - 11 = 64^{\frac{2}{3}}$

Step 2.2

Simplify $64^{\frac{2}{3}}$ .

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Step 2.2.1

Rewrite.

$x - 11 = 0 + 0 + 64^{\frac{2}{3}}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Rewrite $64$ as $4^{3}$ .

$x - 11 = {(4^{3})}^{\frac{2}{3}}$

Step 2.2.2.2

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

$x - 11 = 4^{3 (\frac{2}{3})}$

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

Cancel the common factor.

$x - 11 = 4^{3 (\frac{2}{3})}$

Step 2.2.3.2

Rewrite the expression.

$x - 11 = 4^{2}$

Step 2.2.4

Raise $4$ to the power of $2$ .

$x - 11 = 16$

Step 2.3

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

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Step 2.3.1

Add $11$ to both sides of the equation.

$x = 16 + 11$

Step 2.3.2

Add $16$ and $11$ .

$x = 27$