Solve e^-tfrac{1{100}}=90/100

$$e ^ { - t \frac { 1 } { 100 } } = \frac { 90 } { 100 }$$

Solve for t

$$t=\ln(\frac{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}{265613988875874769338781322035779626829233452653394495974574961739092490901302182994384699044001})\approx 10.536051566$$

Steps Using Definition of Logarithm

Reduce the fraction \frac{90}{100} to lowest terms by extracting and canceling out 10.

$$e^{\left(-t\right)\times \frac{1}{100}}=\frac{9}{10}$$

Multiply -1 and \frac{1}{100} to get -\frac{1}{100}.

$$e^{-\frac{1}{100}t}=\frac{9}{10}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-\frac{1}{100}t})=\log(\frac{9}{10})$$

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

$$-\frac{1}{100}t\log(e)=\log(\frac{9}{10})$$

Divide both sides by \log(e).

$$-\frac{1}{100}t=\frac{\log(\frac{9}{10})}{\log(e)}$$

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

$$-\frac{1}{100}t=\log_{e}\left(\frac{9}{10}\right)$$

Multiply both sides by -100.

$$t=\frac{\ln(\frac{9}{10})}{-\frac{1}{100}}$$