Solve e^lambda-1=0

$${ e }^{ \lambda } -1=0$$

Solve for λ

$$\lambda =0$$

Steps Using Definition of Logarithm

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

$$e^{\lambda }-1=0$$

Add 1 to both sides of the equation.

$$e^{\lambda }=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{\lambda })=\log(1)$$

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

$$\lambda \log(e)=\log(1)$$

Divide both sides by \log(e).

$$\lambda =\frac{\log(1)}{\log(e)}$$

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

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

Solve for λ (complex solution)

$$\lambda =2\pi n_{1}i$$

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