Solve for t A=A_0e^(-0.000121t)

$A = A_{0} e^{- 0.000121 t}$

Rewrite the equation as $A_{0} e^{- 0.000121 t} = A$ .

$A_{0} e^{- 0.000121 t} = A$

Divide each term in $A_{0} e^{- 0.000121 t} = A$ by $A_{0}$ and simplify.

Divide each term in $A_{0} e^{- 0.000121 t} = A$ by $A_{0}$ .

$\frac{A_{0} e^{- 0.000121 t}}{A_{0}} = \frac{A}{A_{0}}$

Simplify the left side.

Cancel the common factor of $A_{0}$ .

Cancel the common factor.

$\frac{A_{0} e^{- 0.000121 t}}{A_{0}} = \frac{A}{A_{0}}$

Divide $e^{- 0.000121 t}$ by $1$ .

$e^{- 0.000121 t} = \frac{A}{A_{0}}$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 0.000121 t}) = \ln (\frac{A}{A_{0}})$

Expand the left side.

Expand $\ln (e^{- 0.000121 t})$ by moving $- 0.000121 t$ outside the logarithm.

$- 0.000121 t \ln (e) = \ln (\frac{A}{A_{0}})$

The natural logarithm of $e$ is $1$ .

$- 0.000121 t \cdot 1 = \ln (\frac{A}{A_{0}})$

Multiply $- 0.000121$ by $1$ .

$- 0.000121 t = \ln (\frac{A}{A_{0}})$

Divide each term in $- 0.000121 t = \ln (\frac{A}{A_{0}})$ by $- 0.000121$ and simplify.

Divide each term in $- 0.000121 t = \ln (\frac{A}{A_{0}})$ by $- 0.000121$ .

$\frac{- 0.000121 t}{- 0.000121} = \frac{\ln (\frac{A}{A_{0}})}{- 0.000121}$

Simplify the left side.

Cancel the common factor of $- 0.000121$ .

Cancel the common factor.

$\frac{- 0.000121 t}{- 0.000121} = \frac{\ln (\frac{A}{A_{0}})}{- 0.000121}$

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{A}{A_{0}})}{- 0.000121}$

Simplify the right side.

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{A}{A_{0}})}{0.000121}$

Multiply by $1$ .

$t = - \frac{1 \ln (\frac{A}{A_{0}})}{0.000121}$

Factor $0.000121$ out of $0.000121$ .

$t = - \frac{1 \ln (\frac{A}{A_{0}})}{0.000121 (1)}$

Separate fractions.

$t = - (\frac{1}{0.000121} \cdot \frac{\ln (\frac{A}{A_{0}})}{1})$

Divide $1$ by $0.000121$ .

$t = - (8264.46280991 \frac{\ln (\frac{A}{A_{0}})}{1})$

Divide $\ln (\frac{A}{A_{0}})$ by $1$ .

$t = - (8264.46280991 \ln (\frac{A}{A_{0}}))$

Multiply $8264.46280991$ by $- 1$ .

$t = - 8264.46280991 \ln (\frac{A}{A_{0}})$