Finite Math Examples

$45000 = 24000 e^{r \cdot 10}$

Step 1

Rewrite the equation as $24000 e^{r \cdot 10} = 45000$ .

$24000 e^{r \cdot 10} = 45000$

Step 2

Divide each term in $24000 e^{r \cdot 10} = 45000$ by $24000$ and simplify.

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Step 2.1

Divide each term in $24000 e^{r \cdot 10} = 45000$ by $24000$ .

$\frac{24000 e^{r \cdot 10}}{24000} = \frac{45000}{24000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $24000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{24000 e^{r \cdot 10}}{24000} = \frac{45000}{24000}$

Step 2.2.1.2

Divide $e^{r \cdot 10}$ by $1$ .

$e^{r \cdot 10} = \frac{45000}{24000}$

Step 2.2.2

Move $10$ to the left of $r$ .

$e^{10 r} = \frac{45000}{24000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $45000$ and $24000$ .

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Step 2.3.1.1

Factor $3000$ out of $45000$ .

$e^{10 r} = \frac{3000 (15)}{24000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $3000$ out of $24000$ .

$e^{10 r} = \frac{3000 \cdot 15}{3000 \cdot 8}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{10 r} = \frac{3000 \cdot 15}{3000 \cdot 8}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{10 r} = \frac{15}{8}$

Step 3

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

$\ln (e^{10 r}) = \ln (\frac{15}{8})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{10 r})$ by moving $10 r$ outside the logarithm.

$10 r \ln (e) = \ln (\frac{15}{8})$

Step 4.2

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

$10 r \cdot 1 = \ln (\frac{15}{8})$

Step 4.3

Multiply $10$ by $1$ .

$10 r = \ln (\frac{15}{8})$

Step 5

Divide each term in $10 r = \ln (\frac{15}{8})$ by $10$ and simplify.

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Step 5.1

Divide each term in $10 r = \ln (\frac{15}{8})$ by $10$ .

$\frac{10 r}{10} = \frac{\ln (\frac{15}{8})}{10}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $10$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{10 r}{10} = \frac{\ln (\frac{15}{8})}{10}$

Step 5.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (\frac{15}{8})}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\ln (\frac{15}{8})}{10}$

Decimal Form:

$r = 0.06286086 \dots$