Solve 23=40e^-0.2h

$$23 = 40 e ^ { - 0.2 h }$$

Solve for h

$$h=\ln(\frac{102400000}{6436343})\approx 2.766926191$$

Steps Using Definition of Logarithm

Divide both sides by 40.

$$\frac{23}{40}=e^{-0.2h}$$

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

$$e^{-0.2h}=\frac{23}{40}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.2h})=\log(\frac{23}{40})$$

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

$$-0.2h\log(e)=\log(\frac{23}{40})$$

Divide both sides by \log(e).

$$-0.2h=\frac{\log(\frac{23}{40})}{\log(e)}$$

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

$$-0.2h=\log_{e}\left(\frac{23}{40}\right)$$

Multiply both sides by -5.

$$h=\frac{\ln(\frac{23}{40})}{-0.2}$$