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 $4$ .

$x = - 1$