Solve 5^y+8=11

$$5 ^ { y + 8 } = 11$$

Solve for y

$$y=\log_{5}\left(11\right)-8\approx -6.510103898$$

Steps Using Definition of Logarithm

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

$$5^{y+8}=11$$

Take the logarithm of both sides of the equation.

$$\log(5^{y+8})=\log(11)$$

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

$$\left(y+8\right)\log(5)=\log(11)$$

Divide both sides by \log(5).

$$y+8=\frac{\log(11)}{\log(5)}$$

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

$$y+8=\log_{5}\left(11\right)$$

Subtract 8 from both sides of the equation.

$$y=\log_{5}\left(11\right)-8$$

Solve for y (complex solution)

$$y=\frac{i\times 2\pi n_{1}}{\ln(5)}+\log_{5}\left(11\right)-8$$

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