Trigonometry Examples

$5 e^{2 t} = 42$

Step 1

Divide each term in $5 e^{2 t} = 42$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 e^{2 t} = 42$ by $5$ .

$\frac{5 e^{2 t}}{5} = \frac{42}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 e^{2 t}}{5} = \frac{42}{5}$

Step 1.2.1.2

Divide $e^{2 t}$ by $1$ .

$e^{2 t} = \frac{42}{5}$

Step 2

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

$\ln (e^{2 t}) = \ln (\frac{42}{5})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{2 t})$ by moving $2 t$ outside the logarithm.

$2 t \ln (e) = \ln (\frac{42}{5})$

Step 3.2

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

$2 t \cdot 1 = \ln (\frac{42}{5})$

Step 3.3

Multiply $2$ by $1$ .

$2 t = \ln (\frac{42}{5})$

Step 4

Divide each term in $2 t = \ln (\frac{42}{5})$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 t = \ln (\frac{42}{5})$ by $2$ .

$\frac{2 t}{2} = \frac{\ln (\frac{42}{5})}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{\ln (\frac{42}{5})}{2}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{42}{5})}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\ln (\frac{42}{5})}{2}$

Decimal Form:

$t = 1.06411585 \dots$