Solve 11^x-8-5=54

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

Solve for x

$$x=\log_{11}\left(59\right)+8\approx 9.700465191$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

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

Add 5 to both sides of the equation.

$$11^{x-8}=59$$

Take the logarithm of both sides of the equation.

$$\log(11^{x-8})=\log(59)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(x-8\right)\log(11)=\log(59)$$

Divide both sides by \log(11).

$$x-8=\frac{\log(59)}{\log(11)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x-8=\log_{11}\left(59\right)$$

Add 8 to both sides of the equation.

$$x=\log_{11}\left(59\right)-\left(-8\right)$$