Solve for n 11^(n-8)-5=54

$11^{n - 8} - 5 = 54$

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

Add $5$ to both sides of the equation.

$11^{n - 8} = 54 + 5$

Add $54$ and $5$ .

$11^{n - 8} = 59$

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

$\ln (11^{n - 8}) = \ln (59)$

Expand $\ln (11^{n - 8})$ by moving $n - 8$ outside the logarithm.

$(n - 8) \ln (11) = \ln (59)$

Simplify the left side.

Apply the distributive property.

$n \ln (11) - 8 \ln (11) = \ln (59)$

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

$n \ln (11) - 8 \ln (11) - \ln (59) = 0$

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

Add $8 \ln (11)$ to both sides of the equation.

$n \ln (11) - \ln (59) = 8 \ln (11)$

Add $\ln (59)$ to both sides of the equation.

$n \ln (11) = 8 \ln (11) + \ln (59)$

Divide each term in $n \ln (11) = 8 \ln (11) + \ln (59)$ by $\ln (11)$ and simplify.

Divide each term in $n \ln (11) = 8 \ln (11) + \ln (59)$ by $\ln (11)$ .

$\frac{n \ln (11)}{\ln (11)} = \frac{8 \ln (11)}{\ln (11)} + \frac{\ln (59)}{\ln (11)}$

Simplify the left side.

Cancel the common factor of $\ln (11)$ .

Cancel the common factor.

$\frac{n \ln (11)}{\ln (11)} = \frac{8 \ln (11)}{\ln (11)} + \frac{\ln (59)}{\ln (11)}$

Divide $n$ by $1$ .

$n = \frac{8 \ln (11)}{\ln (11)} + \frac{\ln (59)}{\ln (11)}$

Simplify the right side.

Cancel the common factor of $\ln (11)$ .

Cancel the common factor.

$n = \frac{8 \ln (11)}{\ln (11)} + \frac{\ln (59)}{\ln (11)}$

Divide $8$ by $1$ .

$n = 8 + \frac{\ln (59)}{\ln (11)}$

The result can be shown in multiple forms.

Exact Form:

$n = 8 + \frac{\ln (59)}{\ln (11)}$

Decimal Form:

$n = 9.70046519 \dots$