Solve 11^x-1-5=54

$$11 ^ { x - 1 } - 5 = 54$$

Solve for x

$$x=\log_{11}\left(59\right)+1\approx 2.700465191$$

Steps Using Definition of Logarithm

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

$$11^{x-1}-5=54$$

Add 5 to both sides of the equation.

$$11^{x-1}=59$$

Take the logarithm of both sides of the equation.

$$\log(11^{x-1})=\log(59)$$

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

$$\left(x-1\right)\log(11)=\log(59)$$

Divide both sides by \log(11).

$$x-1=\frac{\log(59)}{\log(11)}$$

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

$$x-1=\log_{11}\left(59\right)$$

Add 1 to both sides of the equation.

$$x=\log_{11}\left(59\right)-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(11)}+\log_{11}\left(649\right)$$

$$n_{1}\in \mathrm{Z}$$