Solve 3^x=400

$$3 ^ { x } = 400$$

Solve for x

$$x=2\log_{3}\left(20\right)\approx 5.453666056$$

Steps Using Definition of Logarithm

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

$$3^{x}=400$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(400)$$

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

$$x\log(3)=\log(400)$$

Divide both sides by \log(3).

$$x=\frac{\log(400)}{\log(3)}$$

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

$$x=\log_{3}\left(400\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+2\log_{3}\left(20\right)$$

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