Solve 75=100e^0.000121t

$$75 = 100 { e }^{ 0.000121t }$$

Solve for t

$$t=\frac{1000000\ln(3)-2000000\ln(2)}{121}\approx -2377.537788858$$

Steps Using Definition of Logarithm

Divide both sides by 100.

$$\frac{75}{100}=e^{0.000121t}$$

Reduce the fraction \frac{75}{100} to lowest terms by extracting and canceling out 25.

$$\frac{3}{4}=e^{0.000121t}$$

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

$$e^{0.000121t}=\frac{3}{4}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.000121t})=\log(\frac{3}{4})$$

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

$$0.000121t\log(e)=\log(\frac{3}{4})$$

Divide both sides by \log(e).

$$0.000121t=\frac{\log(\frac{3}{4})}{\log(e)}$$

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

$$0.000121t=\log_{e}\left(\frac{3}{4}\right)$$

Divide both sides of the equation by 0.000121, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{3}{4})}{0.000121}$$