Trigonometry Examples

$2 e^{3 x - 1} = 5$

Step 1

Divide each term in $2 e^{3 x - 1} = 5$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 e^{3 x - 1} = 5$ by $2$ .

$\frac{2 e^{3 x - 1}}{2} = \frac{5}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 e^{3 x - 1}}{2} = \frac{5}{2}$

Step 1.2.1.2

Divide $e^{3 x - 1}$ by $1$ .

$e^{3 x - 1} = \frac{5}{2}$

Step 2

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

$\ln (e^{3 x - 1}) = \ln (\frac{5}{2})$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{3 x - 1})$ by moving $3 x - 1$ outside the logarithm.

$(3 x - 1) \ln (e) = \ln (\frac{5}{2})$

Step 3.2

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

$(3 x - 1) \cdot 1 = \ln (\frac{5}{2})$

Step 3.3

Multiply $3 x - 1$ by $1$ .

$3 x - 1 = \ln (\frac{5}{2})$

Step 4

Add $1$ to both sides of the equation.

$3 x = \ln (\frac{5}{2}) + 1$

Step 5

Divide each term in $3 x = \ln (\frac{5}{2}) + 1$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \ln (\frac{5}{2}) + 1$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (\frac{5}{2})}{3} + \frac{1}{3}$

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.

$\frac{3 x}{3} = \frac{\ln (\frac{5}{2})}{3} + \frac{1}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{5}{2})}{3} + \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{5}{2})}{3} + \frac{1}{3}$

Decimal Form:

$x = 0.63876357 \dots$