Precalculus Examples

$\ln (x) = \ln (x + 2) - \ln (2)$

Step 1

Simplify the right side.

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Step 1.1

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

$\ln (x) = \ln (\frac{x + 2}{2})$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $2$ .

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

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

Move $2$ to the left of $x$ .

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

Step 3.2.2

Simplify the right 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.

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

Step 3.2.2.1.2

Rewrite the expression.

$2 x = x + 2$

Step 3.3

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

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Step 3.3.1

Subtract $x$ from both sides of the equation.

$2 x - x = 2$

Step 3.3.2

Subtract $x$ from $2 x$ .

$x = 2$