Algebra Examples

$\log_{3} (4 x) - 2 \log_{3} (x) = 2$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{3} (4 x) - 2 \log_{3} (x) = 2$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\log_{3} (4 x) - 2 \log_{3} (x)$ .

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Step 2.1.1

Simplify $- 2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (4 x) - \log_{3} (x^{2}) = 2$

Step 2.1.2

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

$\log_{3} (\frac{4 x}{x^{2}}) = 2$

Step 2.1.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 2.1.3.1

Factor $x$ out of $4 x$ .

$\log_{3} (\frac{x \cdot 4}{x^{2}}) = 2$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

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

$\log_{3} (\frac{x \cdot 4}{x \cdot x}) = 2$

Step 2.1.3.2.2

Cancel the common factor.

$\log_{3} (\frac{x \cdot 4}{x \cdot x}) = 2$

Step 2.1.3.2.3

Rewrite the expression.

$\log_{3} (\frac{4}{x}) = 2$

Step 3

Rewrite $\log_{3} (\frac{4}{x}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{2} = \frac{4}{x}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{4}{x} = 3^{2}$ .

$\frac{4}{x} = 3^{2}$

Step 4.2

Raise $3$ to the power of $2$ .

$\frac{4}{x} = 9$

Step 4.3

Find the LCD of the terms in the equation.

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Step 4.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 4.3.2

The LCM of one and any expression is the expression.

$x$

Step 4.4

Multiply each term in $\frac{4}{x} = 9$ by $x$ to eliminate the fractions.

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Step 4.4.1

Multiply each term in $\frac{4}{x} = 9$ by $x$ .

$\frac{4}{x} x = 9 x$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $x$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{4}{x} x = 9 x$

Step 4.4.2.1.2

Rewrite the expression.

$4 = 9 x$

Step 4.5

Solve the equation.

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Step 4.5.1

Rewrite the equation as $9 x = 4$ .

$9 x = 4$

Step 4.5.2

Divide each term in $9 x = 4$ by $9$ and simplify.

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Step 4.5.2.1

Divide each term in $9 x = 4$ by $9$ .

$\frac{9 x}{9} = \frac{4}{9}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

Cancel the common factor of $9$ .

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Step 4.5.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{4}{9}$

Step 4.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{9}$

Decimal Form:

$x = 0.\overline{4}$