Finite Math Examples

$90000 = 70797 e^{(0.096) (t)}$

Step 1

Rewrite the equation as $70797 e^{(0.096) (t)} = 90000$ .

$70797 e^{(0.096) (t)} = 90000$

Step 2

Simplify.

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Step 2.1

Multiply $0.096$ by $t$ .

$70797 e^{0.096 (t)} = 90000$

Step 2.2

Multiply $0.096$ by $t$ .

$70797 e^{0.096 t} = 90000$

Step 3

Divide each term in $70797 e^{0.096 t} = 90000$ by $70797$ and simplify.

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Step 3.1

Divide each term in $70797 e^{0.096 t} = 90000$ by $70797$ .

$\frac{70797 e^{0.096 t}}{70797} = \frac{90000}{70797}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $70797$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{70797 e^{0.096 t}}{70797} = \frac{90000}{70797}$

Step 3.2.1.2

Divide $e^{0.096 t}$ by $1$ .

$e^{0.096 t} = \frac{90000}{70797}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $90000$ and $70797$ .

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Step 3.3.1.1

Factor $3$ out of $90000$ .

$e^{0.096 t} = \frac{3 (30000)}{70797}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $3$ out of $70797$ .

$e^{0.096 t} = \frac{3 \cdot 30000}{3 \cdot 23599}$

Step 3.3.1.2.2

Cancel the common factor.

$e^{0.096 t} = \frac{3 \cdot 30000}{3 \cdot 23599}$

Step 3.3.1.2.3

Rewrite the expression.

$e^{0.096 t} = \frac{30000}{23599}$

Step 4

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

$\ln (e^{0.096 t}) = \ln (\frac{30000}{23599})$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{0.096 t})$ by moving $0.096 t$ outside the logarithm.

$0.096 t \ln (e) = \ln (\frac{30000}{23599})$

Step 5.2

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

$0.096 t \cdot 1 = \ln (\frac{30000}{23599})$

Step 5.3

Multiply $0.096$ by $1$ .

$0.096 t = \ln (\frac{30000}{23599})$

Step 6

Divide each term in $0.096 t = \ln (\frac{30000}{23599})$ by $0.096$ and simplify.

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Step 6.1

Divide each term in $0.096 t = \ln (\frac{30000}{23599})$ by $0.096$ .

$\frac{0.096 t}{0.096} = \frac{\ln (\frac{30000}{23599})}{0.096}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $0.096$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{0.096 t}{0.096} = \frac{\ln (\frac{30000}{23599})}{0.096}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{30000}{23599})}{0.096}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Replace $e$ with an approximation.

$t = \frac{\log_{2.71828182} (\frac{30000}{23599})}{0.096}$

Step 6.3.2

Divide $30000$ by $23599$ .

$t = \frac{\log_{2.71828182} (1.2712403)}{0.096}$

Step 6.3.3

Log base $2.71828182$ of $1.2712403$ is approximately $0.23999304$ .

$t = \frac{0.23999304}{0.096}$

Step 6.3.4

Divide $0.23999304$ by $0.096$ .

$t = 2.49992753$