Solve 450=130{(e)^-71(x)}

$$450=130 {(e)^{ -71(x) }}$$

Solve for x

$$x=\frac{\ln(\frac{13}{45})}{71}\approx -0.017488917$$

Steps Using Definition of Logarithm

Divide both sides by 130.

$$\frac{450}{130}=e^{-71x}$$

Reduce the fraction \frac{450}{130} to lowest terms by extracting and canceling out 10.

$$\frac{45}{13}=e^{-71x}$$

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

$$e^{-71x}=\frac{45}{13}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-71x})=\log(\frac{45}{13})$$

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

$$-71x\log(e)=\log(\frac{45}{13})$$

Divide both sides by \log(e).

$$-71x=\frac{\log(\frac{45}{13})}{\log(e)}$$

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

$$-71x=\log_{e}\left(\frac{45}{13}\right)$$

Divide both sides by -71.

$$x=\frac{\ln(\frac{45}{13})}{-71}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{71}+\frac{\ln(\frac{13}{45})}{71}$$

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