Precalculus Examples

$\log_{9} (\frac{9^{6}}{27})$

Step 1

Raise $9$ to the power of $6$ .

$\log_{9} (\frac{531441}{27})$

Step 2

Logarithm base $9$ of $\frac{531441}{27}$ is $\frac{9}{2}$ .

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Step 2.1

Rewrite as an equation.

$\log_{9} (\frac{531441}{27}) = x$

Step 2.2

Rewrite $\log_{9} (\frac{531441}{27}) = x$ 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 $b^{y} = x$ .

$9^{x} = \frac{531441}{27}$

Step 2.3

Create expressions in the equation that all have equal bases.

${(3^{2})}^{x} = 3^{9}$

Step 2.4

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

$3^{2 x} = 3^{9}$

Step 2.5

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

$2 x = 9$

Step 2.6

Solve for $x$ .

$x = \frac{9}{2}$

Step 2.7

The variable $x$ is equal to $\frac{9}{2}$ .

$\frac{9}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$