Calculus Examples

$\log_{2} ({(8)}^{3})$

Step 1

Raise $8$ to the power of $3$ .

$\log_{2} (512)$

Step 2

Logarithm base $2$ of $512$ is $9$ .

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Step 2.1

Rewrite as an equation.

$\log_{2} (512) = x$

Step 2.2

Rewrite $\log_{2} (512) = x$ 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 $b^{y} = x$ .

$2^{x} = 512$

Step 2.3

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{9}$

Step 2.4

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

$x = 9$

Step 2.5

The variable $x$ is equal to $9$ .

$9$