Trigonometry Examples

$18^{- 9 n} = 75$

Step 1

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

$\ln (18^{- 9 n}) = \ln (75)$

Step 2

Expand $\ln (18^{- 9 n})$ by moving $- 9 n$ outside the logarithm.

$- 9 n \ln (18) = \ln (75)$

Step 3

Divide each term in $- 9 n \ln (18) = \ln (75)$ by $- 9 \ln (18)$ and simplify.

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Step 3.1

Divide each term in $- 9 n \ln (18) = \ln (75)$ by $- 9 \ln (18)$ .

$\frac{- 9 n \ln (18)}{- 9 \ln (18)} = \frac{\ln (75)}{- 9 \ln (18)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 9 n \ln (18)}{- 9 \ln (18)} = \frac{\ln (75)}{- 9 \ln (18)}$

Step 3.2.1.2

Rewrite the expression.

$\frac{n \ln (18)}{\ln (18)} = \frac{\ln (75)}{- 9 \ln (18)}$

Step 3.2.2

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

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Step 3.2.2.1

Cancel the common factor.

$\frac{n \ln (18)}{\ln (18)} = \frac{\ln (75)}{- 9 \ln (18)}$

Step 3.2.2.2

Divide $n$ by $1$ .

$n = \frac{\ln (75)}{- 9 \ln (18)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$n = - \frac{\ln (75)}{9 \ln (18)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (75)}{9 \ln (18)}$

Decimal Form:

$n = - 0.16597204 \dots$