Algebra Examples

$\log_{3} (2 x + 3) - \log_{3} (8) = \log_{3} (x)$

Step 1

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

$\log_{3} (\frac{2 x + 3}{8}) = \log_{3} (x)$

Step 2

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

$\frac{2 x + 3}{8} = x$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $8$ .

$\frac{2 x + 3}{8} \cdot 8 = x \cdot 8$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $8$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{2 x + 3}{8} \cdot 8 = x \cdot 8$

Step 3.2.1.1.2

Rewrite the expression.

$2 x + 3 = x \cdot 8$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Move $8$ to the left of $x$ .

$2 x + 3 = 8 x$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1.1

Subtract $8 x$ from both sides of the equation.

$2 x + 3 - 8 x = 0$

Step 3.3.1.2

Subtract $8 x$ from $2 x$ .

$- 6 x + 3 = 0$

Step 3.3.2

Subtract $3$ from both sides of the equation.

$- 6 x = - 3$

Step 3.3.3

Divide each term in $- 6 x = - 3$ by $- 6$ and simplify.

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Step 3.3.3.1

Divide each term in $- 6 x = - 3$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 6}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Cancel the common factor of $- 3$ and $- 6$ .

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Step 3.3.3.3.1.1

Factor $- 3$ out of $- 3$ .

$x = \frac{- 3 (1)}{- 6}$

Step 3.3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.3.1.2.1

Factor $- 3$ out of $- 6$ .

$x = \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Step 3.3.3.3.1.2.2

Cancel the common factor.

$x = \frac{- 3 \cdot 1}{- 3 \cdot 2}$

Step 3.3.3.3.1.2.3

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$