Solve 0.55=e^-0.019254t

$$0.55 = { e }^{ -0.019254t }$$

Solve for t

$$t=\frac{500000\ln(20)-500000\ln(11)}{9627}\approx 31.05001562$$

Steps Using Definition of Logarithm

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

$$e^{-0.019254t}=0.55$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.019254t})=\log(0.55)$$

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

$$-0.019254t\log(e)=\log(0.55)$$

Divide both sides by \log(e).

$$-0.019254t=\frac{\log(0.55)}{\log(e)}$$

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

$$-0.019254t=\log_{e}\left(0.55\right)$$

Divide both sides of the equation by -0.019254, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{11}{20})}{-0.019254}$$