Precalculus Examples

2 e^{3 x} = 1

$2 e^{3 x} = 1$

Step 1

Divide each term in

2 e^{3 x} = 1

$2 e^{3 x} = 1$ by

2

$2$ and simplify.

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Step 1.1

Divide each term in

2 e^{3 x} = 1

$2 e^{3 x} = 1$ by

2

$2$ .

\frac{2 e^{3 x}}{2} = \frac{1}{2}

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

2

$2$ .

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Step 1.2.1.1

Cancel the common factor.

\frac{2 e^{3 x}}{2} = \frac{1}{2}

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

Step 1.2.1.2

Divide

e^{3 x}

$e^{3 x}$ by

1

$1$ .

e^{3 x} = \frac{1}{2}

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

e^{3 x} = \frac{1}{2}

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

e^{3 x} = \frac{1}{2}

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

e^{3 x} = \frac{1}{2}

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

Step 2

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

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

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

Step 3

Expand the left side.

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Step 3.1

Expand

\ln (e^{3 x})

$\ln (e^{3 x})$ by moving

3 x

$3 x$ outside the logarithm.

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

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

Step 3.2

The natural logarithm of

e

$e$ is

1

$1$ .

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

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

Step 3.3

Multiply

3

$3$ by

1

$1$ .

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

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

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

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

Step 4

Divide each term in

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

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

3

$3$ and simplify.

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Step 4.1

Divide each term in

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

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

3

$3$ .

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

3

$3$ .

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Step 4.2.1.1

Cancel the common factor.

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

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

Step 4.2.1.2

Divide

x

$x$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

x = - 0.23104906 \dots

$x = - 0.23104906 \dots$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$