Finite Math Examples

$50000 = 28000 e^{6 r}$

Step 1

Rewrite the equation as $28000 e^{6 r} = 50000$ .

$28000 e^{6 r} = 50000$

Step 2

Divide each term in $28000 e^{6 r} = 50000$ by $28000$ and simplify.

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Step 2.1

Divide each term in $28000 e^{6 r} = 50000$ by $28000$ .

$\frac{28000 e^{6 r}}{28000} = \frac{50000}{28000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $28000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{28000 e^{6 r}}{28000} = \frac{50000}{28000}$

Step 2.2.1.2

Divide $e^{6 r}$ by $1$ .

$e^{6 r} = \frac{50000}{28000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $50000$ and $28000$ .

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Step 2.3.1.1

Factor $2000$ out of $50000$ .

$e^{6 r} = \frac{2000 (25)}{28000}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2000$ out of $28000$ .

$e^{6 r} = \frac{2000 \cdot 25}{2000 \cdot 14}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{6 r} = \frac{2000 \cdot 25}{2000 \cdot 14}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{6 r} = \frac{25}{14}$

Step 3

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

$\ln (e^{6 r}) = \ln (\frac{25}{14})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{6 r})$ by moving $6 r$ outside the logarithm.

$6 r \ln (e) = \ln (\frac{25}{14})$

Step 4.2

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

$6 r \cdot 1 = \ln (\frac{25}{14})$

Step 4.3

Multiply $6$ by $1$ .

$6 r = \ln (\frac{25}{14})$

Step 5

Divide each term in $6 r = \ln (\frac{25}{14})$ by $6$ and simplify.

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Step 5.1

Divide each term in $6 r = \ln (\frac{25}{14})$ by $6$ .

$\frac{6 r}{6} = \frac{\ln (\frac{25}{14})}{6}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $6$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{6 r}{6} = \frac{\ln (\frac{25}{14})}{6}$

Step 5.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (\frac{25}{14})}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\ln (\frac{25}{14})}{6}$

Decimal Form:

$r = 0.09663641 \dots$