Solve 4^4x-6=34

$$4 ^ { 4 x - 6 } = 34$$

Solve for x

$$x = \frac{\log_{2} {(34)} + 12}{8} \approx 2.135932855$$

Steps Using Definition of Logarithm

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

$$4^{4x-6}=34$$

Take the logarithm of both sides of the equation.

$$\log(4^{4x-6})=\log(34)$$

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

$$\left(4x-6\right)\log(4)=\log(34)$$

Divide both sides by \log(4).

$$4x-6=\frac{\log(34)}{\log(4)}$$

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

$$4x-6=\log_{4}\left(34\right)$$

Add 6 to both sides of the equation.

$$4x=\frac{\log_{2}\left(34\right)}{2}-\left(-6\right)$$

Divide both sides by 4.

$$x=\frac{\frac{\log_{2}\left(34\right)}{2}+6}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{4\ln(2)}+\frac{\log_{2}\left(17\right)}{8}+\frac{13}{8}$$

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