Solve e^-10x=2

$$e ^ { - 10 x } = 2$$

Solve for x

$$x=-\frac{\ln(2)}{10}\approx -0.069314718$$

Steps Using Definition of Logarithm

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

$$e^{-10x}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{-10x})=\log(2)$$

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

$$-10x\log(e)=\log(2)$$

Divide both sides by \log(e).

$$-10x=\frac{\log(2)}{\log(e)}$$

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

$$-10x=\log_{e}\left(2\right)$$

Divide both sides by -10.

$$x=\frac{\ln(2)}{-10}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{5}-\frac{\ln(2)}{10}$$

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