Solve 7^2y=12

$$7 ^ { 2 y } = 12$$

Solve for y

$$y=\frac{\log_{7}\left(12\right)}{2}\approx 0.638494704$$

Steps Using Definition of Logarithm

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

$$7^{2y}=12$$

Take the logarithm of both sides of the equation.

$$\log(7^{2y})=\log(12)$$

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

$$2y\log(7)=\log(12)$$

Divide both sides by \log(7).

$$2y=\frac{\log(12)}{\log(7)}$$

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

$$2y=\log_{7}\left(12\right)$$

Divide both sides by 2.

$$y=\frac{\log_{7}\left(12\right)}{2}$$

Solve for y (complex solution)

$$y=\frac{\pi n_{1}i}{\ln(7)}+\frac{\log_{7}\left(12\right)}{2}$$

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