Finite Math Examples

$81000 = 59871 e^{(0.093) t}$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$59871 e^{(0.093) t} = 81000$

Step 2

Multiply $0.093$ by $t$ .

$59871 e^{0.093 t} = 81000$

Step 3

Divide each term in $59871 e^{0.093 t} = 81000$ by $59871$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $59871 e^{0.093 t} = 81000$ by $59871$ .

$\frac{59871 e^{0.093 t}}{59871} = \frac{81000}{59871}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $59871$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{59871 e^{0.093 t}}{59871} = \frac{81000}{59871}$

Step 3.2.1.2

Divide $e^{0.093 t}$ by $1$ .

$e^{0.093 t} = \frac{81000}{59871}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $81000$ and $59871$ .

Tap for more steps...

Step 3.3.1.1

Factor $3$ out of $81000$ .

$e^{0.093 t} = \frac{3 (27000)}{59871}$

Step 3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.1.2.1

Factor $3$ out of $59871$ .

$e^{0.093 t} = \frac{3 \cdot 27000}{3 \cdot 19957}$

Step 3.3.1.2.2

Cancel the common factor.

$e^{0.093 t} = \frac{3 \cdot 27000}{3 \cdot 19957}$

Step 3.3.1.2.3

Rewrite the expression.

$e^{0.093 t} = \frac{27000}{19957}$

Step 4

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

$\ln (e^{0.093 t}) = \ln (\frac{27000}{19957})$

Step 5

Expand the left side.

Tap for more steps...

Step 5.1

Expand $\ln (e^{0.093 t})$ by moving $0.093 t$ outside the logarithm.

$0.093 t \ln (e) = \ln (\frac{27000}{19957})$

Step 5.2

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

$0.093 t \cdot 1 = \ln (\frac{27000}{19957})$

Step 5.3

Multiply $0.093$ by $1$ .

$0.093 t = \ln (\frac{27000}{19957})$

Step 6

Divide each term in $0.093 t = \ln (\frac{27000}{19957})$ by $0.093$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $0.093 t = \ln (\frac{27000}{19957})$ by $0.093$ .

$\frac{0.093 t}{0.093} = \frac{\ln (\frac{27000}{19957})}{0.093}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $0.093$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{0.093 t}{0.093} = \frac{\ln (\frac{27000}{19957})}{0.093}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{27000}{19957})}{0.093}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{27000}{19957})}{0.093}$

Step 6.3.2

Divide $27000$ by $19957$ .

$t = \frac{\log_{2.71828182} (1.35290875)}{0.093}$

Step 6.3.3

Log base $2.71828182$ of $1.35290875$ is approximately $0.3022569$ .

$t = \frac{0.3022569}{0.093}$

Step 6.3.4

Divide $0.3022569$ by $0.093$ .

$t = 3.25007426$