Solve 4^2y-3=102

$$4 ^ { 2 y - 3 } = 102$$

Solve for y

$$y = \frac{\log_{2} {(102)} + 6}{4} \approx 3.168106335$$

Steps Using Definition of Logarithm

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

$$4^{2y-3}=102$$

Take the logarithm of both sides of the equation.

$$\log(4^{2y-3})=\log(102)$$

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

$$\left(2y-3\right)\log(4)=\log(102)$$

Divide both sides by \log(4).

$$2y-3=\frac{\log(102)}{\log(4)}$$

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

$$2y-3=\log_{4}\left(102\right)$$

Add 3 to both sides of the equation.

$$2y=\frac{\log_{2}\left(102\right)}{2}-\left(-3\right)$$

Divide both sides by 2.

$$y=\frac{\frac{\log_{2}\left(102\right)}{2}+3}{2}$$