Precalculus Examples

$10000 = 8600 e^{r}$

Step 1

Rewrite the equation as $8600 e^{r} = 10000$ .

$8600 e^{r} = 10000$

Step 2

Divide each term in $8600 e^{r} = 10000$ by $8600$ and simplify.

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Step 2.1

Divide each term in $8600 e^{r} = 10000$ by $8600$ .

$\frac{8600 e^{r}}{8600} = \frac{10000}{8600}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8600$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8600 e^{r}}{8600} = \frac{10000}{8600}$

Step 2.2.1.2

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

$e^{r} = \frac{10000}{8600}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $10000$ and $8600$ .

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Step 2.3.1.1

Factor $200$ out of $10000$ .

$e^{r} = \frac{200 (50)}{8600}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $200$ out of $8600$ .

$e^{r} = \frac{200 \cdot 50}{200 \cdot 43}$

Step 2.3.1.2.2

Cancel the common factor.

$e^{r} = \frac{200 \cdot 50}{200 \cdot 43}$

Step 2.3.1.2.3

Rewrite the expression.

$e^{r} = \frac{50}{43}$

Step 3

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

$\ln (e^{r}) = \ln (\frac{50}{43})$

Step 4

Expand the left side.

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Step 4.1

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

$r \ln (e) = \ln (\frac{50}{43})$

Step 4.2

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

$r \cdot 1 = \ln (\frac{50}{43})$

Step 4.3

Multiply $r$ by $1$ .

$r = \ln (\frac{50}{43})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = \ln (\frac{50}{43})$

Decimal Form:

$r = 0.15082288 \dots$