Solve e^alpha=10

$$e ^ { \alpha } = 10$$

Solve for α

$$\alpha =\ln(10)\approx 2.302585093$$

Steps Using Definition of Logarithm

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

$$e^{\alpha }=10$$

Take the logarithm of both sides of the equation.

$$\log(e^{\alpha })=\log(10)$$

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

$$\alpha \log(e)=\log(10)$$

Divide both sides by \log(e).

$$\alpha =\frac{\log(10)}{\log(e)}$$

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

$$\alpha =\log_{e}\left(10\right)$$

Solve for α (complex solution)

$$\alpha =\ln(10)+2\pi n_{1}i$$

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