Solve e^-0.0174z=0.5

$${ e }^{ -0.0174z } =0.5$$

Solve for z

$$z=\frac{5000\ln(2)}{87}\approx 39.83604486$$

Steps Using Definition of Logarithm

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

$$e^{-0.0174z}=0.5$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.0174z})=\log(0.5)$$

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

$$-0.0174z\log(e)=\log(0.5)$$

Divide both sides by \log(e).

$$-0.0174z=\frac{\log(0.5)}{\log(e)}$$

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

$$-0.0174z=\log_{e}\left(0.5\right)$$

Divide both sides of the equation by -0.0174, which is the same as multiplying both sides by the reciprocal of the fraction.

$$z=-\frac{\ln(2)}{-0.0174}$$

Solve for z (complex solution)

$$z=-\frac{i\times 10000\pi n_{1}}{87}+\frac{5000\ln(2)}{87}$$

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