Solve 22.073e^-0.066x=20

$$22.073 e ^ { - 0.066 x } = 20$$

Solve for x

$$x=\frac{500\ln(22073)-500\ln(20000)}{33}\approx 1.494285888$$

Steps Using Definition of Logarithm

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

$$22.073e^{-0.066x}=20$$

Divide both sides of the equation by 22.073, which is the same as multiplying both sides by the reciprocal of the fraction.

$$e^{-0.066x}=\frac{20000}{22073}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.066x})=\log(\frac{20000}{22073})$$

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

$$-0.066x\log(e)=\log(\frac{20000}{22073})$$

Divide both sides by \log(e).

$$-0.066x=\frac{\log(\frac{20000}{22073})}{\log(e)}$$

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

$$-0.066x=\log_{e}\left(\frac{20000}{22073}\right)$$

Divide both sides of the equation by -0.066, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{20000}{22073})}{-0.066}$$

Solve for x (complex solution)

$$x=-\frac{i\times 1000\pi n_{1}}{33}+\frac{500\ln(22073)}{33}-\frac{500\ln(20000)}{33}$$

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