Precalculus Examples

$\log (6 x + 5) - \log (2) = \log (5)$

Step 1

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

$\log (\frac{6 x + 5}{2}) = \log (5)$

Step 2

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

$\frac{6 x + 5}{2} = 5$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{6 x + 5}{2} \cdot 2 = 5 \cdot 2$

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.

$\frac{6 x + 5}{2} \cdot 2 = 5 \cdot 2$

Step 3.2.1.1.2

Rewrite the expression.

$6 x + 5 = 5 \cdot 2$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $5$ by $2$ .

$6 x + 5 = 10$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $5$ from both sides of the equation.

$6 x = 10 - 5$

Step 3.3.1.2

Subtract $5$ from $10$ .

$6 x = 5$

Step 3.3.2

Divide each term in $6 x = 5$ by $6$ and simplify.

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Step 3.3.2.1

Divide each term in $6 x = 5$ by $6$ .

$\frac{6 x}{6} = \frac{5}{6}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{5}{6}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{6}$

Decimal Form:

$x = 0.8\overline{3}$