Solve for n e^(8.5n)-4=41

$e^{8.5 n} - 4 = 41$

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

Add $4$ to both sides of the equation.

$e^{8.5 n} = 41 + 4$

Add $41$ and $4$ .

$e^{8.5 n} = 45$

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

$\ln (e^{8.5 n}) = \ln (45)$

Expand the left side.

Expand $\ln (e^{8.5 n})$ by moving $8.5 n$ outside the logarithm.

$8.5 n \ln (e) = \ln (45)$

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

$8.5 n \cdot 1 = \ln (45)$

Multiply $8.5$ by $1$ .

$8.5 n = \ln (45)$

Divide each term in $8.5 n = \ln (45)$ by $8.5$ and simplify.

Divide each term in $8.5 n = \ln (45)$ by $8.5$ .

$\frac{8.5 n}{8.5} = \frac{\ln (45)}{8.5}$

Simplify the left side.

Cancel the common factor of $8.5$ .

Cancel the common factor.

$\frac{8.5 n}{8.5} = \frac{\ln (45)}{8.5}$

Divide $n$ by $1$ .

$n = \frac{\ln (45)}{8.5}$

Simplify the right side.

Replace $e$ with an approximation.

$n = \frac{\log_{2.71828182} (45)}{8.5}$

Log base $2.71828182$ of $45$ is approximately $3.80666248$ .

$n = \frac{3.80666248}{8.5}$

Divide $3.80666248$ by $8.5$ .

$n = 0.44784264$