Algebra Examples

$x + 5 = \log_{4} (256)$

Step 1

Logarithm base

$4$ of

$256$ is

$4$ .

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Step 1.1

Rewrite as an equation.

$x + 5 = \log_{4} (256) = x$

Step 1.2

Rewrite

$\log_{4} (256) = x$ 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

$b^{y} = x$ .

$x + 5 = 4^{x} = 256$

Step 1.3

Create expressions in the equation that all have equal bases.

$x + 5 = {(2^{2})}^{x} = 2^{8}$

Step 1.4

Rewrite

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

$2^{2 x}$ .

$x + 5 = 2^{2 x} = 2^{8}$

Step 1.5

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

$x + 5 = 2 x = 8$

Step 1.6

Solve for

$x$ .

$x + 5 = x = 4$

Step 1.7

The variable

$x$ is equal to

$4$ .

$x + 5 = 4$

Step 2

Move all terms not containing

$x$ to the right side of the equation.

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Step 2.1

Subtract

$5$ from both sides of the equation.

$x = 4 - 5$

Step 2.2

Subtract

$5$ from

$4$ .

$x = - 1$