Solve e^-2x=4

$$e ^ { - 2 x } = 4$$

Solve for x

$$x=-\ln(2)\approx -0.693147181$$

Steps Using Definition of Logarithm

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

$$e^{-2x}=4$$

Take the logarithm of both sides of the equation.

$$\log(e^{-2x})=\log(4)$$

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

$$-2x\log(e)=\log(4)$$

Divide both sides by \log(e).

$$-2x=\frac{\log(4)}{\log(e)}$$

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

$$-2x=\log_{e}\left(4\right)$$

Divide both sides by -2.

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

Solve for x (complex solution)

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

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