Solve e^-3y=9

$$e ^ { - 3 y } = 9$$

Solve for y

$$y=-\frac{2\ln(3)}{3}\approx -0.732408192$$

Steps Using Definition of Logarithm

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

$$e^{-3y}=9$$

Take the logarithm of both sides of the equation.

$$\log(e^{-3y})=\log(9)$$

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

$$-3y\log(e)=\log(9)$$

Divide both sides by \log(e).

$$-3y=\frac{\log(9)}{\log(e)}$$

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

$$-3y=\log_{e}\left(9\right)$$

Divide both sides by -3.

$$y=\frac{2\ln(3)}{-3}$$

Solve for y (complex solution)

$$y=-\frac{i\times 2\pi n_{1}}{3}-\frac{2\ln(3)}{3}$$

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