Trigonometry Examples

$8 \log_{9} (- 4 n) = 24$

Step 1

Divide each term in $8 \log_{9} (- 4 n) = 24$ by $8$ and simplify.

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Step 1.1

Divide each term in $8 \log_{9} (- 4 n) = 24$ by $8$ .

$\frac{8 \log_{9} (- 4 n)}{8} = \frac{24}{8}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $8$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{8 \log_{9} (- 4 n)}{8} = \frac{24}{8}$

Step 1.2.1.2

Divide $\log_{9} (- 4 n)$ by $1$ .

$\log_{9} (- 4 n) = \frac{24}{8}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $24$ by $8$ .

$\log_{9} (- 4 n) = 3$

Step 2

Rewrite $\log_{9} (- 4 n) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$9^{3} = - 4 n$

Step 3

Solve for $n$ .

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Step 3.1

Rewrite the equation as $- 4 n = 9^{3}$ .

$- 4 n = 9^{3}$

Step 3.2

Divide each term in $- 4 n = 9^{3}$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 n = 9^{3}$ by $- 4$ .

$\frac{- 4 n}{- 4} = \frac{9^{3}}{- 4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 n}{- 4} = \frac{9^{3}}{- 4}$

Step 3.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{9^{3}}{- 4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raise $9$ to the power of $3$ .

$n = \frac{729}{- 4}$

Step 3.2.3.2

Move the negative in front of the fraction.

$n = - \frac{729}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{729}{4}$

Decimal Form:

$n = - 182.25$

Mixed Number Form:

$n = - 182 \frac{1}{4}$