Solve e^-9y=11

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

Solve for y

$$y=-\frac{\ln(11)}{9}\approx -0.266432808$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

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

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

$$-9y\log(e)=\log(11)$$

Divide both sides by \log(e).

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

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

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

Divide both sides by -9.

$$y=\frac{\ln(11)}{-9}$$

Solve for y (complex solution)

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

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