Solve 12^-5y=2

$$12 ^ { - 5 y } = 2$$

Solve for y

$$y=-\frac{\log_{12}\left(2\right)}{5}\approx -0.055788589$$

Steps Using Definition of Logarithm

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

$$12^{-5y}=2$$

Take the logarithm of both sides of the equation.

$$\log(12^{-5y})=\log(2)$$

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

$$-5y\log(12)=\log(2)$$

Divide both sides by \log(12).

$$-5y=\frac{\log(2)}{\log(12)}$$

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

$$-5y=\log_{12}\left(2\right)$$

Divide both sides by -5.

$$y=\frac{\log_{12}\left(2\right)}{-5}$$

Solve for y (complex solution)

$$y=-\frac{2\pi n_{1}i}{5\ln(12)}-\frac{\log_{12}\left(2\right)}{5}$$

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