Algebra Examples

$\log_{3} (18 x^{3}) - \log_{3} (2 x) = \log_{3} (144)$

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{18 x^{3}}{2 x}) = \log_{3} (144)$

Step 1.2

Cancel the common factor of $18$ and $2$ .

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Step 1.2.1

Factor $2$ out of $18 x^{3}$ .

$\log_{3} (\frac{2 (9 x^{3})}{2 x}) = \log_{3} (144)$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $2 x$ .

$\log_{3} (\frac{2 (9 x^{3})}{2 (x)}) = \log_{3} (144)$

Step 1.2.2.2

Cancel the common factor.

$\log_{3} (\frac{2 (9 x^{3})}{2 x}) = \log_{3} (144)$

Step 1.2.2.3

Rewrite the expression.

$\log_{3} (\frac{9 x^{3}}{x}) = \log_{3} (144)$

Step 1.3

Cancel the common factor of $x^{3}$ and $x$ .

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Step 1.3.1

Factor $x$ out of $9 x^{3}$ .

$\log_{3} (\frac{x (9 x^{2})}{x}) = \log_{3} (144)$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Raise $x$ to the power of $1$ .

$\log_{3} (\frac{x (9 x^{2})}{x}) = \log_{3} (144)$

Step 1.3.2.2

Factor $x$ out of $x^{1}$ .

$\log_{3} (\frac{x (9 x^{2})}{x \cdot 1}) = \log_{3} (144)$

Step 1.3.2.3

Cancel the common factor.

$\log_{3} (\frac{x (9 x^{2})}{x \cdot 1}) = \log_{3} (144)$

Step 1.3.2.4

Rewrite the expression.

$\log_{3} (\frac{9 x^{2}}{1}) = \log_{3} (144)$

Step 1.3.2.5

Divide $9 x^{2}$ by $1$ .

$\log_{3} (9 x^{2}) = \log_{3} (144)$

Step 2

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

$9 x^{2} = 144$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $9 x^{2} = 144$ by $9$ and simplify.

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Step 3.1.1

Divide each term in $9 x^{2} = 144$ by $9$ .

$\frac{9 x^{2}}{9} = \frac{144}{9}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} = \frac{144}{9}$

Step 3.1.2.1.2

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

$x^{2} = \frac{144}{9}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $144$ by $9$ .

$x^{2} = 16$

Step 3.2

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

$x = \pm \sqrt{16}$

Step 3.3

Simplify $\pm \sqrt{16}$ .

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Step 3.3.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 3.3.2

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

$x = \pm 4$

Step 3.4

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

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Step 3.4.1

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

$x = 4$

Step 3.4.2

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

$x = - 4$

Step 3.4.3

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

$x = 4, - 4$

Step 4

Exclude the solutions that do not make $\log_{3} (18 x^{3}) - \log_{3} (2 x) = \log_{3} (144)$ true.

$x = 4$