Solve e^xpi=1

$${ e }^{ x \pi } =1$$

Solve for x

$$x=0$$

Steps Using Definition of Logarithm

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

$$e^{\pi x}=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{\pi x})=\log(1)$$

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

$$\pi x\log(e)=\log(1)$$

Divide both sides by \log(e).

$$\pi x=\frac{\log(1)}{\log(e)}$$

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

$$\pi x=\log_{e}\left(1\right)$$

Divide both sides by \pi .

$$x=\frac{0}{\pi }$$

Solve for x (complex solution)

$$x=2n_{1}i$$

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