Finite Math Examples

$1200 = 500 e^{(r) (9)}$

Step 1

Rewrite the equation as $500 e^{(r) (9)} = 1200$ .

$500 e^{(r) (9)} = 1200$

Step 2

Multiply $r$ by $9$ .

$500 e^{r (9)} = 1200$

Step 3

Divide each term in $500 e^{r (9)} = 1200$ by $500$ and simplify.

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Step 3.1

Divide each term in $500 e^{r (9)} = 1200$ by $500$ .

$\frac{500 e^{r (9)}}{500} = \frac{1200}{500}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $500$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{500 e^{r (9)}}{500} = \frac{1200}{500}$

Step 3.2.1.2

Divide $e^{r (9)}$ by $1$ .

$e^{r (9)} = \frac{1200}{500}$

Step 3.2.2

Move $9$ to the left of $r$ .

$e^{9 r} = \frac{1200}{500}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $1200$ and $500$ .

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Step 3.3.1.1

Factor $100$ out of $1200$ .

$e^{9 r} = \frac{100 (12)}{500}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $100$ out of $500$ .

$e^{9 r} = \frac{100 \cdot 12}{100 \cdot 5}$

Step 3.3.1.2.2

Cancel the common factor.

$e^{9 r} = \frac{100 \cdot 12}{100 \cdot 5}$

Step 3.3.1.2.3

Rewrite the expression.

$e^{9 r} = \frac{12}{5}$

Step 4

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

$\ln (e^{9 r}) = \ln (\frac{12}{5})$

Step 5

Expand the left side.

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Step 5.1

Expand $\ln (e^{9 r})$ by moving $9 r$ outside the logarithm.

$9 r \ln (e) = \ln (\frac{12}{5})$

Step 5.2

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

$9 r \cdot 1 = \ln (\frac{12}{5})$

Step 5.3

Multiply $9$ by $1$ .

$9 r = \ln (\frac{12}{5})$

Step 6

Divide each term in $9 r = \ln (\frac{12}{5})$ by $9$ and simplify.

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Step 6.1

Divide each term in $9 r = \ln (\frac{12}{5})$ by $9$ .

$\frac{9 r}{9} = \frac{\ln (\frac{12}{5})}{9}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $9$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{9 r}{9} = \frac{\ln (\frac{12}{5})}{9}$

Step 6.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (\frac{12}{5})}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\ln (\frac{12}{5})}{9}$

Decimal Form:

$r = 0.09727430 \dots$