Calculus Examples

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

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

Step 1

Raise

8

$8$ to the power of

3

$3$ .

\log_{2} (512)

$\log_{2} (512)$

Step 2

Logarithm base

2

$2$ of

512

$512$ is

9

$9$ .

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Step 2.1

Rewrite as an equation.

\log_{2} (512) = x

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

Step 2.2

Rewrite

\log_{2} (512) = x

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

$b^{y} = x$ .

2^{x} = 512

$2^{x} = 512$

Step 2.3

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

2^{x} = 2^{9}

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

Step 2.4

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

x = 9

$x = 9$

Step 2.5

The variable

x

$x$ is equal to

9

$9$ .

9

$9$

9

$9$