Solve e^y+9=11

$$e ^ { y + 9 } = 11$$

Solve for y

$$y=\ln(11)-9\approx -6.602104727$$

Steps Using Definition of Logarithm

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

$$e^{y+9}=11$$

Take the logarithm of both sides of the equation.

$$\log(e^{y+9})=\log(11)$$

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

$$\left(y+9\right)\log(e)=\log(11)$$

Divide both sides by \log(e).

$$y+9=\frac{\log(11)}{\log(e)}$$

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

$$y+9=\log_{e}\left(11\right)$$

Subtract 9 from both sides of the equation.

$$y=\ln(11)-9$$

Solve for y (complex solution)

$$y=\ln(11)-9+2\pi n_{1}i$$

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