Algebra Examples

{log}_{4} (2)

$\log_{4} (2)$

Step 1

Rewrite as an equation.

{log}_{4} (2) = x

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

Step 2

Rewrite

{log}_{4} (2) = x

$\log_{4} (2) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$ does not equal

1

$1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

4^{x} = 2

$4^{x} = 2$

Step 3

Create expressions in the equation that all have equal bases.

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

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

Step 4

Rewrite

{(2^{2})}^{x}

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

2^{2 x}

$2^{2 x}$ .

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

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

Step 5

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

2 x = 1

$2 x = 1$

Step 6

Solve for

x

$x$ .

x = \frac{1}{2}

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

Step 7

The variable

x

$x$ is equal to

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2}

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

\frac{1}{2}

$\frac{1}{2}$

Decimal Form:

0.5

$0.5$