Finite Math Examples

$200 e^{\frac{t}{4}} = 1500$

Step 1

Divide each term in $200 e^{\frac{t}{4}} = 1500$ by $200$ and simplify.

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Step 1.1

Divide each term in $200 e^{\frac{t}{4}} = 1500$ by $200$ .

$\frac{200 e^{\frac{t}{4}}}{200} = \frac{1500}{200}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $200$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{200 e^{\frac{t}{4}}}{200} = \frac{1500}{200}$

Step 1.2.1.2

Divide $e^{\frac{t}{4}}$ by $1$ .

$e^{\frac{t}{4}} = \frac{1500}{200}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $1500$ and $200$ .

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Step 1.3.1.1

Factor $100$ out of $1500$ .

$e^{\frac{t}{4}} = \frac{100 (15)}{200}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $100$ out of $200$ .

$e^{\frac{t}{4}} = \frac{100 \cdot 15}{100 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

$e^{\frac{t}{4}} = \frac{100 \cdot 15}{100 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

$e^{\frac{t}{4}} = \frac{15}{2}$

Step 2

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

$\ln (e^{\frac{t}{4}}) = \ln (\frac{15}{2})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{\frac{t}{4}})$ by moving $\frac{t}{4}$ outside the logarithm.

$\frac{t}{4} \ln (e) = \ln (\frac{15}{2})$

Step 3.2

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

$\frac{t}{4} \cdot 1 = \ln (\frac{15}{2})$

Step 3.3

Multiply $\frac{t}{4}$ by $1$ .

$\frac{t}{4} = \ln (\frac{15}{2})$

Step 4

Multiply both sides of the equation by $4$ .

$4 \frac{t}{4} = 4 \ln (\frac{15}{2})$

Step 5

Simplify the left side.

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Step 5.1

Cancel the common factor of $4$ .

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Step 5.1.1

Cancel the common factor.

$4 \frac{t}{4} = 4 \ln (\frac{15}{2})$

Step 5.1.2

Rewrite the expression.

$t = 4 \ln (\frac{15}{2})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = 4 \ln (\frac{15}{2})$

Decimal Form:

$t = 8.05961208 \dots$