Precalculus Examples

$9 \cdot 18^{2 - 8 n} + 9 = 39$

Move all terms not containing $n$ to the right side of the equation.

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Subtract $9$ from both sides of the equation.

$9 \cdot 18^{2 - 8 n} = 39 - 9$

Subtract $9$ from $39$ .

$9 \cdot 18^{2 - 8 n} = 30$

Divide each term in $(9 \cdot 18^{2 - 8 n}) = 30$ by $9$ and simplify.

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Divide each term in $(9 \cdot 18^{2 - 8 n}) = 30$ by $9$ .

$\frac{(9 \cdot 18^{2 - 8 n})}{9} = \frac{30}{9}$

Simplify the left side.

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Cancel the common factor of $9$ .

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Cancel the common factor.

$\frac{9 \cdot 18^{2 - 8 n}}{9} = \frac{30}{9}$

Divide $18^{2 - 8 n}$ by $1$ .

$18^{2 - 8 n} = \frac{30}{9}$

Simplify the right side.

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Cancel the common factor of $30$ and $9$ .

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Factor $3$ out of $30$ .

$18^{2 - 8 n} = \frac{3 (10)}{9}$

Cancel the common factors.

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Factor $3$ out of $9$ .

$18^{2 - 8 n} = \frac{3 \cdot 10}{3 \cdot 3}$

Cancel the common factor.

$18^{2 - 8 n} = \frac{3 \cdot 10}{3 \cdot 3}$

Rewrite the expression.

$18^{2 - 8 n} = \frac{10}{3}$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (18^{2 - 8 n}) = \ln (\frac{10}{3})$

Expand $\ln (18^{2 - 8 n})$ by moving $2 - 8 n$ outside the logarithm.

$(2 - 8 n) \ln (18) = \ln (\frac{10}{3})$

Simplify the left side.

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Apply the distributive property.

$2 \ln (18) - 8 n \ln (18) = \ln (\frac{10}{3})$

Reorder $2 \ln (18)$ and $- 8 n \ln (18)$ .

$- 8 n \ln (18) + 2 \ln (18) = \ln (\frac{10}{3})$

Move all the terms containing a logarithm to the left side of the equation.

$- 8 n \ln (18) + 2 \ln (18) - \ln (\frac{10}{3}) = 0$

Move all terms not containing $n$ to the right side of the equation.

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Subtract $2 \ln (18)$ from both sides of the equation.

$- 8 n \ln (18) - \ln (\frac{10}{3}) = - 2 \ln (18)$

Add $\ln (\frac{10}{3})$ to both sides of the equation.

$- 8 n \ln (18) = - 2 \ln (18) + \ln (\frac{10}{3})$

Divide each term in $- 8 n \ln (18) = - 2 \ln (18) + \ln (\frac{10}{3})$ by $- 8 \ln (18)$ and simplify.

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Divide each term in $- 8 n \ln (18) = - 2 \ln (18) + \ln (\frac{10}{3})$ by $- 8 \ln (18)$ .

$\frac{- 8 n \ln (18)}{- 8 \ln (18)} = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Simplify the left side.

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Cancel the common factor of $- 8$ .

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Cancel the common factor.

$\frac{- 8 n \ln (18)}{- 8 \ln (18)} = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Rewrite the expression.

$\frac{n \ln (18)}{\ln (18)} = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Cancel the common factor of $\ln (18)$ .

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Cancel the common factor.

$\frac{n \ln (18)}{\ln (18)} = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Divide $n$ by $1$ .

$n = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Simplify the right side.

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Simplify each term.

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Cancel the common factor of $- 2$ and $- 8$ .

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Factor $- 2$ out of $- 2 \ln (18)$ .

$n = \frac{- 2 \ln (18)}{- 8 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Cancel the common factors.

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Factor $- 2$ out of $- 8 \ln (18)$ .

$n = \frac{- 2 \ln (18)}{- 2 (4 \ln (18))} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Cancel the common factor.

$n = \frac{- 2 \ln (18)}{- 2 (4 \ln (18))} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Rewrite the expression.

$n = \frac{\ln (18)}{4 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Cancel the common factor of $\ln (18)$ .

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Cancel the common factor.

$n = \frac{\ln (18)}{4 \ln (18)} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Rewrite the expression.

$n = \frac{1}{4} + \frac{\ln (\frac{10}{3})}{- 8 \ln (18)}$

Move the negative in front of the fraction.

$n = \frac{1}{4} - \frac{\ln (\frac{10}{3})}{8 \ln (18)}$

The result can be shown in multiple forms.

Exact Form:

$n = \frac{1}{4} - \frac{\ln (\frac{10}{3})}{8 \ln (18)}$

Decimal Form:

$n = 0.19793174 \dots$