Algebra Examples

$\log_{6} (2 x - 3) = \log_{6} (12) - \log_{6} (3)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log_{6} (12) - \log_{6} (3)$ .

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Step 1.1.1

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

$\log_{6} (2 x - 3) = \log_{6} (\frac{12}{3})$

Step 1.1.2

Divide $12$ by $3$ .

$\log_{6} (2 x - 3) = \log_{6} (4)$

Step 2

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

$2 x - 3 = 4$

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 $3$ to both sides of the equation.

$2 x = 4 + 3$

Step 3.1.2

Add $4$ and $3$ .

$2 x = 7$

Step 3.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{2}$

Decimal Form:

$x = 3.5$

Mixed Number Form:

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