Algebra Examples

$\ln (x^{2} - 9) + \ln (9) = \ln (40)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((x^{2} - 9) \cdot 9) = \ln (40)$

Step 1.2

Apply the distributive property.

$\ln (x^{2} \cdot 9 - 9 \cdot 9) = \ln (40)$

Step 1.3

Simplify the expression.

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Step 1.3.1

Move $9$ to the left of $x^{2}$ .

$\ln (9 \cdot x^{2} - 9 \cdot 9) = \ln (40)$

Step 1.3.2

Multiply $- 9$ by $9$ .

$\ln (9 x^{2} - 81) = \ln (40)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$9 x^{2} - 81 = 40$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Add $81$ to both sides of the equation.

$9 x^{2} = 40 + 81$

Step 3.1.2

Add $40$ and $81$ .

$9 x^{2} = 121$

Step 3.2

Divide each term in $9 x^{2} = 121$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 x^{2} = 121$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{121}{9}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{121}{9}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{121}{9}$

Step 3.3

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

$x = \pm \sqrt{\frac{121}{9}}$

Step 3.4

Simplify $\pm \sqrt{\frac{121}{9}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{121}{9}}$ as $\frac{\sqrt{121}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{121}}{\sqrt{9}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Rewrite $121$ as $11^{2}$ .

$x = \pm \frac{\sqrt{11^{2}}}{\sqrt{9}}$

Step 3.4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{11}{\sqrt{9}}$

Step 3.4.3

Simplify the denominator.

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Step 3.4.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{11}{\sqrt{3^{2}}}$

Step 3.4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{11}{3}$

Step 3.5

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

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Step 3.5.1

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

$x = \frac{11}{3}$

Step 3.5.2

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

$x = - \frac{11}{3}$

Step 3.5.3

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

$x = \frac{11}{3}, - \frac{11}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}, - \frac{11}{3}$

Decimal Form:

$x = 3.\overline{6}, - 3.\overline{6}$

Mixed Number Form:

$x = 3 \frac{2}{3}, - 3 \frac{2}{3}$