Basic Math Examples

1000 {(e)}^{0.09 t} = 5000

$1000 {(e)}^{0.09 t} = 5000$

Step 1

Divide each term in

1000 e^{0.09 t} = 5000

$1000 e^{0.09 t} = 5000$ by

1000

$1000$ and simplify.

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Step 1.1

Divide each term in

1000 e^{0.09 t} = 5000

$1000 e^{0.09 t} = 5000$ by

1000

$1000$ .

\frac{1000 e^{0.09 t}}{1000} = \frac{5000}{1000}

$\frac{1000 e^{0.09 t}}{1000} = \frac{5000}{1000}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

1000

$1000$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{1000 e^{0.09 t}}{1000} = \frac{5000}{1000}$

Step 1.2.1.2

Divide

$e^{0.09 t}$ by

$1$ .

$e^{0.09 t} = \frac{5000}{1000}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide

$5000$ by

$1000$ .

$e^{0.09 t} = 5$

Step 2

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

$\ln (e^{0.09 t}) = \ln (5)$

Step 3

Expand the left side.

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Step 3.1

Expand

$\ln (e^{0.09 t})$ by moving

$0.09 t$ outside the logarithm.

$0.09 t \ln (e) = \ln (5)$

Step 3.2

The natural logarithm of

$e$ is

$1$ .

$0.09 t \cdot 1 = \ln (5)$

Step 3.3

Multiply

$0.09$ by

$1$ .

$0.09 t = \ln (5)$

Step 4

Divide each term in

$0.09 t = \ln (5)$ by

$0.09$ and simplify.

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Step 4.1

Divide each term in

$0.09 t = \ln (5)$ by

$0.09$ .

$\frac{0.09 t}{0.09} = \frac{\ln (5)}{0.09}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

$0.09$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.09 t}{0.09} = \frac{\ln (5)}{0.09}$

Step 4.2.1.2

Divide

$t$ by

$1$ .

$t = \frac{\ln (5)}{0.09}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace

$e$ with an approximation.

$t = \frac{\log_{2.71828182} (5)}{0.09}$

Step 4.3.2

Log base

$2.71828182$ of

$5$ is approximately

$1.60943791$ .

$t = \frac{1.60943791}{0.09}$

Step 4.3.3

Divide

$1.60943791$ by

$0.09$ .

$t = 17.88264347$