Solve e^-6y=15

$$e ^ { - 6 y } = 15$$

Solve for y

$$y=-\frac{\ln(15)}{6}\approx -0.4513417$$

Steps Using Definition of Logarithm

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

$$e^{-6y}=15$$

Take the logarithm of both sides of the equation.

$$\log(e^{-6y})=\log(15)$$

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

$$-6y\log(e)=\log(15)$$

Divide both sides by \log(e).

$$-6y=\frac{\log(15)}{\log(e)}$$

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

$$-6y=\log_{e}\left(15\right)$$

Divide both sides by -6.

$$y=\frac{\ln(15)}{-6}$$

Solve for y (complex solution)

$$y=-\frac{i\pi n_{1}}{3}-\frac{\ln(15)}{6}$$

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