Solve for n e^(-8n-9)=36.5

$e^{- 8 n - 9} = 36.5$

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

$\ln (e^{- 8 n - 9}) = \ln (36.5)$

Expand the left side.

Expand $\ln (e^{- 8 n - 9})$ by moving $- 8 n - 9$ outside the logarithm.

$(- 8 n - 9) \ln (e) = \ln (36.5)$

The natural logarithm of $e$ is $1$ .

$(- 8 n - 9) \cdot 1 = \ln (36.5)$

Multiply $- 8 n - 9$ by $1$ .

$- 8 n - 9 = \ln (36.5)$

Add $9$ to both sides of the equation.

$- 8 n = \ln (36.5) + 9$

Divide each term in $- 8 n = \ln (36.5) + 9$ by $- 8$ and simplify.

Divide each term in $- 8 n = \ln (36.5) + 9$ by $- 8$ .

$\frac{- 8 n}{- 8} = \frac{\ln (36.5)}{- 8} + \frac{9}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 n}{- 8} = \frac{\ln (36.5)}{- 8} + \frac{9}{- 8}$

Divide $n$ by $1$ .

$n = \frac{\ln (36.5)}{- 8} + \frac{9}{- 8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$n = - \frac{\ln (36.5)}{8} + \frac{9}{- 8}$

Move the negative in front of the fraction.

$n = - \frac{\ln (36.5)}{8} - \frac{9}{8}$

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (36.5)}{8} - \frac{9}{8}$

Decimal Form:

$n = - 1.57466403 \dots$