Solve e^-0.66t=0.37

$$e ^ { - 0.66 t } = 0.37$$

Solve for t

$$t=\frac{100\ln(10)-50\ln(37)}{33}\approx 1.506442838$$

Steps Using Definition of Logarithm

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

$$e^{-0.66t}=0.37$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.66t})=\log(0.37)$$

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

$$-0.66t\log(e)=\log(0.37)$$

Divide both sides by \log(e).

$$-0.66t=\frac{\log(0.37)}{\log(e)}$$

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

$$-0.66t=\log_{e}\left(0.37\right)$$

Divide both sides of the equation by -0.66, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{37}{100})}{-0.66}$$