Basic Math Examples

$30.2 = 0.02165 e^{r (43)}$

Step 1

Rewrite the equation as $0.02165 e^{r (43)} = 30.2$ .

$0.02165 e^{r (43)} = 30.2$

Step 2

Divide each term in $0.02165 e^{r (43)} = 30.2$ by $0.02165$ and simplify.

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Step 2.1

Divide each term in $0.02165 e^{r (43)} = 30.2$ by $0.02165$ .

$\frac{0.02165 e^{r (43)}}{0.02165} = \frac{30.2}{0.02165}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.02165$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.02165 e^{r (43)}}{0.02165} = \frac{30.2}{0.02165}$

Step 2.2.1.2

Divide $e^{r (43)}$ by $1$ .

$e^{r (43)} = \frac{30.2}{0.02165}$

Step 2.2.2

Move $43$ to the left of $r$ .

$e^{43 r} = \frac{30.2}{0.02165}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $30.2$ by $0.02165$ .

$e^{43 r} = 1394.91916859$

Step 3

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

$\ln (e^{43 r}) = \ln (1394.91916859)$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{43 r})$ by moving $43 r$ outside the logarithm.

$43 r \ln (e) = \ln (1394.91916859)$

Step 4.2

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

$43 r \cdot 1 = \ln (1394.91916859)$

Step 4.3

Multiply $43$ by $1$ .

$43 r = \ln (1394.91916859)$

Step 5

Divide each term in $43 r = \ln (1394.91916859)$ by $43$ and simplify.

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Step 5.1

Divide each term in $43 r = \ln (1394.91916859)$ by $43$ .

$\frac{43 r}{43} = \frac{\ln (1394.91916859)}{43}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $43$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{43 r}{43} = \frac{\ln (1394.91916859)}{43}$

Step 5.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (1394.91916859)}{43}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\ln (1394.91916859)}{43}$

Decimal Form:

$r = 0.16838585 \dots$