Solve 6^y-2=11

$$6 ^ { y - 2 } = 11$$

Solve for y

$$y=\log_{6}\left(11\right)+2\approx 3.338290833$$

Steps Using Definition of Logarithm

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

$$6^{y-2}=11$$

Take the logarithm of both sides of the equation.

$$\log(6^{y-2})=\log(11)$$

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

$$\left(y-2\right)\log(6)=\log(11)$$

Divide both sides by \log(6).

$$y-2=\frac{\log(11)}{\log(6)}$$

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

$$y-2=\log_{6}\left(11\right)$$

Add 2 to both sides of the equation.

$$y=\log_{6}\left(11\right)-\left(-2\right)$$

Solve for y (complex solution)

$$y=\frac{i\times 2\pi n_{1}}{\ln(6)}+\log_{6}\left(11\right)+2$$

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