Solve 0.59=e^-0.154t

$$0.59 = e ^ { - 0.154 t }$$

Solve for t

$$t=\frac{1000\ln(10)-500\ln(59)}{77}\approx 3.426186637$$

Steps Using Definition of Logarithm

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

$$e^{-0.154t}=0.59$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.154t})=\log(0.59)$$

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

$$-0.154t\log(e)=\log(0.59)$$

Divide both sides by \log(e).

$$-0.154t=\frac{\log(0.59)}{\log(e)}$$

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

$$-0.154t=\log_{e}\left(0.59\right)$$

Divide both sides of the equation by -0.154, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{59}{100})}{-0.154}$$