Trigonometry Examples

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

Step 1

Write in exponential form.

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Step 1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = {(3 x - 2)}^{2}$ , and $y = 5$ .

$b = e$

$x = {(3 x - 2)}^{2}$

$y = 5$

Step 1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$e^{5} = {(3 x - 2)}^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as ${(3 x - 2)}^{2} = e^{5}$ .

${(3 x - 2)}^{2} = e^{5}$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$3 x - 2 = \pm \sqrt{e^{5}}$

Step 2.3

Simplify $\pm \sqrt{e^{5}}$ .

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Step 2.3.1

Rewrite $e^{5}$ as ${(e^{2})}^{2} e$ .

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Step 2.3.1.1

Factor out $e^{4}$ .

$3 x - 2 = \pm \sqrt{e^{4} e}$

Step 2.3.1.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

$3 x - 2 = \pm \sqrt{{(e^{2})}^{2} e}$

Step 2.3.2

Pull terms out from under the radical.

$3 x - 2 = \pm e^{2} \sqrt{e}$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$3 x - 2 = e^{2} \sqrt{e}$

Step 2.4.2

Add $2$ to both sides of the equation.

$3 x = e^{2} \sqrt{e} + 2$

Step 2.4.3

Divide each term in $3 x = e^{2} \sqrt{e} + 2$ by $3$ and simplify.

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Step 2.4.3.1

Divide each term in $3 x = e^{2} \sqrt{e} + 2$ by $3$ .

$\frac{3 x}{3} = \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $3$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.4

Next, use the negative value of the $\pm$ to find the second solution.

$3 x - 2 = - e^{2} \sqrt{e}$

Step 2.4.5

Add $2$ to both sides of the equation.

$3 x = - e^{2} \sqrt{e} + 2$

Step 2.4.6

Divide each term in $3 x = - e^{2} \sqrt{e} + 2$ by $3$ and simplify.

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Step 2.4.6.1

Divide each term in $3 x = - e^{2} \sqrt{e} + 2$ by $3$ .

$\frac{3 x}{3} = \frac{- e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.6.2

Simplify the left side.

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Step 2.4.6.2.1

Cancel the common factor of $3$ .

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Step 2.4.6.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.6.3

Simplify the right side.

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Step 2.4.6.3.1

Move the negative in front of the fraction.

$x = - \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 2.4.7

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}, - \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}, - \frac{e^{2} \sqrt{e}}{3} + \frac{2}{3}$

Decimal Form:

$x = 4.72749798 \dots, - 3.39416465 \dots$