Solve 4^x=214

$$4 ^ { x } = 214$$

Solve for x

$$x = \frac{\log_{2} {(214)}}{2} \approx 3.870733493$$

Steps Using Definition of Logarithm

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

$$4^{x}=214$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(214)$$

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

$$x\log(4)=\log(214)$$

Divide both sides by \log(4).

$$x=\frac{\log(214)}{\log(4)}$$

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

$$x=\log_{4}\left(214\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(2)}+\frac{\log_{2}\left(214\right)}{2}$$

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