Solve 4^n=5

$$4 ^ { n } = 5$$

Solve for n

$$n = \frac{\log_{2} {(5)}}{2} \approx 1.160964047$$

Steps Using Definition of Logarithm

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

$$4^{n}=5$$

Take the logarithm of both sides of the equation.

$$\log(4^{n})=\log(5)$$

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

$$n\log(4)=\log(5)$$

Divide both sides by \log(4).

$$n=\frac{\log(5)}{\log(4)}$$

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

$$n=\log_{4}\left(5\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{\ln(2)}+\frac{\log_{2}\left(5\right)}{2}$$

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