Solve 30=75e^-0.002t

$$30 = 75 e ^ { - 0.002 t }$$

Solve for t

$$t=\ln(9332636185032188789900895447238171696170914463717080246217143397959669109757756344544403270978811023595949899303242426242154875213540323948415208172039307562344106661383251502739950759859018315111004.90796265113118240512514795933790805178271125415103810698378854426481119469814228660959222017662910442798456169448887147466528006328368452647429261829862165202793195289493607117850663668741065439805530718136320599844826041954101213229629869502194514609904214608668361244792952034826864617657926916047420065936389041737895822118365078045556628444273925387517127854796781556346403714877681766899855392069265439424008711973674701749862626690747296762535803929376233833981046927874558605253696441650390625)$$

Steps Using Definition of Logarithm

Divide both sides by 75.

$$\frac{30}{75}=e^{-0.002t}$$

Reduce the fraction \frac{30}{75} to lowest terms by extracting and canceling out 15.

$$\frac{2}{5}=e^{-0.002t}$$

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

$$e^{-0.002t}=\frac{2}{5}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.002t})=\log(\frac{2}{5})$$

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

$$-0.002t\log(e)=\log(\frac{2}{5})$$

Divide both sides by \log(e).

$$-0.002t=\frac{\log(\frac{2}{5})}{\log(e)}$$

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

$$-0.002t=\log_{e}\left(\frac{2}{5}\right)$$

Multiply both sides by -500.

$$t=\frac{\ln(\frac{2}{5})}{-0.002}$$