Precalculus Examples

$\log_{4} (\sqrt{64}) = x$

Step 1

Rewrite the equation as $x = \log_{4} (\sqrt{64})$ .

$x = \log_{4} (\sqrt{64})$

Step 2

Simplify $\log_{4} (\sqrt{64})$ .

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Step 2.1

Rewrite $64$ as $8^{2}$ .

$x = \log_{4} (\sqrt{8^{2}})$

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Rewrite as an equation.

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

Step 2.3.2

Rewrite $\log_{4} (8) = x$ 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 $b^{y} = x$ .

$x = 4^{x} = 8$

Step 2.3.3

Create expressions in the equation that all have equal bases.

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

Step 2.3.4

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

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

Step 2.3.5

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

$x = 2 x = 3$

Step 2.3.6

Solve for $x$ .

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

Step 2.3.7

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}$