Solve 4^6x-2=45

$$4 ^ { 6 x - 2 } = 45$$

Solve for x

$$x=\frac{\log_{2}\left(45\right)}{12}+\frac{1}{3}\approx 0.790987758$$

Steps Using Definition of Logarithm

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

$$4^{6x-2}=45$$

Take the logarithm of both sides of the equation.

$$\log(4^{6x-2})=\log(45)$$

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

$$\left(6x-2\right)\log(4)=\log(45)$$

Divide both sides by \log(4).

$$6x-2=\frac{\log(45)}{\log(4)}$$

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

$$6x-2=\log_{4}\left(45\right)$$

Add 2 to both sides of the equation.

$$6x=\frac{\log_{2}\left(45\right)}{2}-\left(-2\right)$$

Divide both sides by 6.

$$x=\frac{\frac{\log_{2}\left(45\right)}{2}+2}{6}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{6\ln(2)}+\frac{\log_{2}\left(45\right)}{12}+\frac{1}{3}$$

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