Precalculus Examples

$\log (x + 6) - \log (2) = \log (3 x + 1)$

Step 1

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

$\log (\frac{x + 6}{2}) = \log (3 x + 1)$

Step 2

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

$\frac{x + 6}{2} = 3 x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{x + 6}{2} \cdot 2 = (3 x + 1) \cdot 2$

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 $2$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{x + 6}{2} \cdot 2 = (3 x + 1) \cdot 2$

Step 3.2.1.1.2

Rewrite the expression.

$x + 6 = (3 x + 1) \cdot 2$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(3 x + 1) \cdot 2$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x + 6 = 3 x \cdot 2 + 1 \cdot 2$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $2$ by $3$ .

$x + 6 = 6 x + 1 \cdot 2$

Step 3.2.2.1.2.2

Multiply $2$ by $1$ .

$x + 6 = 6 x + 2$

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 $6 x$ from both sides of the equation.

$x + 6 - 6 x = 2$

Step 3.3.1.2

Subtract $6 x$ from $x$ .

$- 5 x + 6 = 2$

Step 3.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.2.1

Subtract $6$ from both sides of the equation.

$- 5 x = 2 - 6$

Step 3.3.2.2

Subtract $6$ from $2$ .

$- 5 x = - 4$

Step 3.3.3

Divide each term in $- 5 x = - 4$ by $- 5$ and simplify.

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Step 3.3.3.1

Divide each term in $- 5 x = - 4$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 4}{- 5}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 5}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$