Solve 4^x-1=7

$$4 ^ { x - 1 } = 7$$

Solve for x

$$x = \frac{\log_{2} {(7)} + 2}{2} \approx 2.403677461$$

Steps Using Definition of Logarithm

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

$$4^{x-1}=7$$

Take the logarithm of both sides of the equation.

$$\log(4^{x-1})=\log(7)$$

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

$$\left(x-1\right)\log(4)=\log(7)$$

Divide both sides by \log(4).

$$x-1=\frac{\log(7)}{\log(4)}$$

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

$$x-1=\log_{4}\left(7\right)$$

Add 1 to both sides of the equation.

$$x=\frac{\log_{2}\left(7\right)}{2}-\left(-1\right)$$

Solve for x (complex solution)

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

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