Solve 50=200e^-0.014x

$$50 = 200 { e }^{ -0.014x }$$

Solve for x

$$x=\frac{1000\ln(2)}{7}\approx 99.021025794$$

Steps Using Definition of Logarithm

Divide both sides by 200.

$$\frac{50}{200}=e^{-0.014x}$$

Reduce the fraction \frac{50}{200} to lowest terms by extracting and canceling out 50.

$$\frac{1}{4}=e^{-0.014x}$$

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

$$e^{-0.014x}=\frac{1}{4}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.014x})=\log(\frac{1}{4})$$

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

$$-0.014x\log(e)=\log(\frac{1}{4})$$

Divide both sides by \log(e).

$$-0.014x=\frac{\log(\frac{1}{4})}{\log(e)}$$

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

$$-0.014x=\log_{e}\left(\frac{1}{4}\right)$$

Divide both sides of the equation by -0.014, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{2\ln(2)}{-0.014}$$

Solve for x (complex solution)

$$x=-\frac{i\times 1000\pi n_{1}}{7}+\frac{1000\ln(2)}{7}$$

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