Calculus Examples

$\log_{4} (\sqrt{4})$

Step 1

Rewrite $4$ as $2^{2}$ .

$\log_{4} (\sqrt{2^{2}})$

Step 2

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

$\log_{4} (2)$

Step 3

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

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Step 3.1

Rewrite as an equation.

$\log_{4} (2) = x$

Step 3.2

Rewrite $\log_{4} (2) = x$ 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 $b^{y} = x$ .

$4^{x} = 2$

Step 3.3

Create expressions in the equation that all have equal bases.

${(2^{2})}^{x} = 2^{1}$

Step 3.4

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

$2^{2 x} = 2^{1}$

Step 3.5

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

$2 x = 1$

Step 3.6

Solve for $x$ .

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

Step 3.7

The variable $x$ is equal to $\frac{1}{2}$ .

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$