Solve e^x-1=2

$$e ^ { x - 1 } = 2$$

Solve for x

$$x=\ln(2)+1\approx 1.693147181$$

Steps Using Definition of Logarithm

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

$$e^{x-1}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{x-1})=\log(2)$$

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

$$\left(x-1\right)\log(e)=\log(2)$$

Divide both sides by \log(e).

$$x-1=\frac{\log(2)}{\log(e)}$$

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

$$x-1=\log_{e}\left(2\right)$$

Add 1 to both sides of the equation.

$$x=\ln(2)-\left(-1\right)$$

Solve for x (complex solution)

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

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