Solve 0.1=20e^-0.4x

$$0.1 = 20 e ^ { - 0.4 x }$$

Solve for x

$$x=\frac{5\ln(200)}{2}\approx 13.245793416$$

Steps Using Definition of Logarithm

Divide both sides by 20.

$$\frac{0.1}{20}=e^{-0.4x}$$

Expand \frac{0.1}{20} by multiplying both numerator and the denominator by 10.

$$\frac{1}{200}=e^{-0.4x}$$

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

$$e^{-0.4x}=\frac{1}{200}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.4x})=\log(\frac{1}{200})$$

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

$$-0.4x\log(e)=\log(\frac{1}{200})$$

Divide both sides by \log(e).

$$-0.4x=\frac{\log(\frac{1}{200})}{\log(e)}$$

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

$$-0.4x=\log_{e}\left(\frac{1}{200}\right)$$

Divide both sides of the equation by -0.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{\ln(200)}{-0.4}$$

Solve for x (complex solution)

$$x=-i\times 5\pi n_{1}+\frac{5\ln(200)}{2}$$

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