Algebra Examples

$\log_{3} (6 x) - \log_{3} (9) = 4$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{3} (\frac{6 x}{9}) = 4$

Step 1.2

Reduce the expression $\frac{6 x}{9}$ by cancelling the common factors.

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Step 1.2.1

Factor $3$ out of $6 x$ .

$\log_{3} (\frac{3 (2 x)}{9}) = 4$

Step 1.2.2

Factor $3$ out of $9$ .

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

Step 1.2.3

Cancel the common factor.

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

Step 1.2.4

Rewrite the expression.

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

Step 1.3

Rewrite $\frac{2 x}{3}$ as $2 x \cdot 3^{- 1}$ .

$\log_{3} (2 x \cdot 3^{- 1}) = 4$

Step 1.4

Rewrite $\log_{3} (2 x \cdot 3^{- 1})$ as $\log_{3} (2 x) + \log_{3} (3^{- 1})$ .

$\log_{3} (2 x) + \log_{3} (3^{- 1}) = 4$

Step 1.5

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 1.6

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (2 x) - 1 \cdot 1 = 4$

Step 1.7

Multiply $- 1$ by $1$ .

$\log_{3} (2 x) - 1 = 4$

Step 2

Move all terms not containing $\log_{3} (2 x)$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$\log_{3} (2 x) = 4 + 1$

Step 2.2

Add $4$ and $1$ .

$\log_{3} (2 x) = 5$

Step 3

Rewrite $\log_{3} (2 x) = 5$ 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 $b^{y} = x$ .

$3^{5} = 2 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $2 x = 3^{5}$ .

$2 x = 3^{5}$

Step 4.2

Divide each term in $2 x = 3^{5}$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = 3^{5}$ by $2$ .

$\frac{2 x}{2} = \frac{3^{5}}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3^{5}}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3^{5}}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $3$ to the power of $5$ .

$x = \frac{243}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{243}{2}$

Decimal Form:

$x = 121.5$

Mixed Number Form:

$x = 121 \frac{1}{2}$