Solve 31=89.976e^-0.023t

$$31 = 89.976 e ^ { - 0.023 t }$$

Solve for t

$$t=\frac{1000\ln(11247)-1000\ln(3875)}{23}\approx 46.328511462$$

Steps Using Definition of Logarithm

Divide both sides by 89.976.

$$\frac{31}{89.976}=e^{-0.023t}$$

Expand \frac{31}{89.976} by multiplying both numerator and the denominator by 1000.

$$\frac{31000}{89976}=e^{-0.023t}$$

Reduce the fraction \frac{31000}{89976} to lowest terms by extracting and canceling out 8.

$$\frac{3875}{11247}=e^{-0.023t}$$

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

$$e^{-0.023t}=\frac{3875}{11247}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.023t})=\log(\frac{3875}{11247})$$

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

$$-0.023t\log(e)=\log(\frac{3875}{11247})$$

Divide both sides by \log(e).

$$-0.023t=\frac{\log(\frac{3875}{11247})}{\log(e)}$$

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

$$-0.023t=\log_{e}\left(\frac{3875}{11247}\right)$$

Divide both sides of the equation by -0.023, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{3875}{11247})}{-0.023}$$

Solve for t (complex solution)

$$t=-\frac{i\times 2000\pi n_{1}}{23}+\frac{1000\ln(11247)}{23}-\frac{1000\ln(3875)}{23}$$

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