Solve 20=50e^frac{10{3}x}

$$20 = 50 e ^ { \frac { 10 } { 3 } x }$$

Solve for x

$$x=\frac{3\ln(\frac{2}{5})}{10}\approx -0.27488722$$

Steps Using Definition of Logarithm

Divide both sides by 50.

$$\frac{20}{50}=e^{\frac{10}{3}x}$$

Reduce the fraction \frac{20}{50} to lowest terms by extracting and canceling out 10.

$$\frac{2}{5}=e^{\frac{10}{3}x}$$

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

$$e^{\frac{10}{3}x}=\frac{2}{5}$$

Take the logarithm of both sides of the equation.

$$\log(e^{\frac{10}{3}x})=\log(\frac{2}{5})$$

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

$$\frac{10}{3}x\log(e)=\log(\frac{2}{5})$$

Divide both sides by \log(e).

$$\frac{10}{3}x=\frac{\log(\frac{2}{5})}{\log(e)}$$

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

$$\frac{10}{3}x=\log_{e}\left(\frac{2}{5}\right)$$

Divide both sides of the equation by \frac{10}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{2}{5})}{\frac{10}{3}}$$

Solve for x (complex solution)

$$x=\frac{3\pi n_{1}i}{5}+\frac{3\ln(\frac{2}{5})}{10}$$

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