Solve e^x-2=3

$$e ^ { x - 2 } = 3$$

Solve for x

$$x=\ln(3)+2\approx 3.098612289$$

Steps Using Definition of Logarithm

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

$$e^{x-2}=3$$

Take the logarithm of both sides of the equation.

$$\log(e^{x-2})=\log(3)$$

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

$$\left(x-2\right)\log(e)=\log(3)$$

Divide both sides by \log(e).

$$x-2=\frac{\log(3)}{\log(e)}$$

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

$$x-2=\log_{e}\left(3\right)$$

Add 2 to both sides of the equation.

$$x=\ln(3)-\left(-2\right)$$

Solve for x (complex solution)

$$x=\ln(3)+2+2\pi n_{1}i$$

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