Precalculus Examples

$\log_{8} (\sqrt[3]{64})$

Step 1

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

$\log_{8} (\sqrt[3]{4^{3}})$

Step 2

Pull terms out from under the radical, assuming real numbers.

$\log_{8} (4)$

Step 3

Logarithm base $8$ of $4$ is $\frac{2}{3}$ .

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Step 3.1

Rewrite as an equation.

$\log_{8} (4) = x$

Step 3.2

Rewrite $\log_{8} (4) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$8^{x} = 4$

Step 3.3

Create expressions in the equation that all have equal bases.

${(2^{3})}^{x} = 2^{2}$

Step 3.4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

$2^{3 x} = 2^{2}$

Step 3.5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 x = 2$

Step 3.6

Solve for $x$ .

$x = \frac{2}{3}$

Step 3.7

The variable $x$ is equal to $\frac{2}{3}$ .

$\frac{2}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$