Finite Math Examples

$4 = \log_{9} (6561)$

Step 1

Logarithm base $9$ of $6561$ is $4$ .

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Step 1.1

Rewrite as an equation.

$4 = \log_{9} (6561) = x$

Step 1.2

Rewrite $\log_{9} (6561) = x$ 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 $b^{y} = x$ .

$4 = 9^{x} = 6561$

Step 1.3

Create expressions in the equation that all have equal bases.

$4 = {(3^{2})}^{x} = 3^{8}$

Step 1.4

Rewrite ${(3^{2})}^{x}$ as $3^{2 x}$ .

$4 = 3^{2 x} = 3^{8}$

Step 1.5

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

$4 = 2 x = 8$

Step 1.6

Solve for $x$ .

$4 = x = 4$

Step 1.7

The variable $x$ is equal to $4$ .

$4 = 4$

Step 2

Since $4 = 4$ , the equation will always be true.

Always true

Step 3

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$