Solve 75=150e^13k

$$75 = 150 e ^ { 13 k }$$

Solve for k

$$k=-\frac{\ln(2)}{13}\approx -0.053319014$$

Steps Using Definition of Logarithm

Divide both sides by 150.

$$\frac{75}{150}=e^{13k}$$

Reduce the fraction \frac{75}{150} to lowest terms by extracting and canceling out 75.

$$\frac{1}{2}=e^{13k}$$

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

$$e^{13k}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{13k})=\log(\frac{1}{2})$$

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

$$13k\log(e)=\log(\frac{1}{2})$$

Divide both sides by \log(e).

$$13k=\frac{\log(\frac{1}{2})}{\log(e)}$$

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

$$13k=\log_{e}\left(\frac{1}{2}\right)$$

Divide both sides by 13.

$$k=-\frac{\ln(2)}{13}$$

Solve for k (complex solution)

$$k=\frac{2\pi n_{1}i}{13}-\frac{\ln(2)}{13}$$

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