Trigonometry Examples

$11983 = 21018 e^{3 r}$

Step 1

Rewrite the equation as $21018 e^{3 r} = 11983$ .

$21018 e^{3 r} = 11983$

Step 2

Divide each term in $21018 e^{3 r} = 11983$ by $21018$ and simplify.

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Step 2.1

Divide each term in $21018 e^{3 r} = 11983$ by $21018$ .

$\frac{21018 e^{3 r}}{21018} = \frac{11983}{21018}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $21018$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{21018 e^{3 r}}{21018} = \frac{11983}{21018}$

Step 2.2.1.2

Divide $e^{3 r}$ by $1$ .

$e^{3 r} = \frac{11983}{21018}$

Step 3

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

$\ln (e^{3 r}) = \ln (\frac{11983}{21018})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln (e^{3 r})$ by moving $3 r$ outside the logarithm.

$3 r \ln (e) = \ln (\frac{11983}{21018})$

Step 4.2

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

$3 r \cdot 1 = \ln (\frac{11983}{21018})$

Step 4.3

Multiply $3$ by $1$ .

$3 r = \ln (\frac{11983}{21018})$

Step 5

Divide each term in $3 r = \ln (\frac{11983}{21018})$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 r = \ln (\frac{11983}{21018})$ by $3$ .

$\frac{3 r}{3} = \frac{\ln (\frac{11983}{21018})}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 r}{3} = \frac{\ln (\frac{11983}{21018})}{3}$

Step 5.2.1.2

Divide $r$ by $1$ .

$r = \frac{\ln (\frac{11983}{21018})}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\ln (\frac{11983}{21018})}{3}$

Decimal Form:

$r = - 0.18729674 \dots$