Solve 2=e^-2s

$$2 = e ^ { - 2 s }$$

Solve for s

$$s=-\frac{\ln(2)}{2}\approx -0.34657359$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$e^{-2s}=2$$

Take the logarithm of both sides of the equation.

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

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

$$-2s\log(e)=\log(2)$$

Divide both sides by \log(e).

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

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

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

Divide both sides by -2.

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

Solve for s (complex solution)

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

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