Calculus Examples

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

Step 1

Logarithm base $2$ of $8$ is $3$ .

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Step 1.1

Rewrite as an equation.

$\log_{2} (8) = x = 3$

Step 1.2

Rewrite $\log_{2} (8) = x$ 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 $b^{y} = x$ .

$2^{x} = 8 = 3$

Step 1.3

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

$2^{x} = 2^{3} = 3$

Step 1.4

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

$x = 3 = 3$

Step 1.5

The variable $x$ is equal to $3$ .

$3 = 3$

Step 2

Since $3 = 3$ , the equation will always be true.

Always true

Step 3

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$