Precalculus Examples

$\log (x + 4) - \log (x) = \log (x + 2)$

Step 1

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

$\log (\frac{x + 4}{x}) = \log (x + 2)$

Step 2

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

$\frac{x + 4}{x} = x + 2$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$x$

Step 3.2

Multiply each term in $\frac{x + 4}{x} = x + 2$ by $x$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x + 4}{x} = x + 2$ by $x$ .

$\frac{x + 4}{x} x = x \cdot x + 2 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{x + 4}{x} x = x \cdot x + 2 x$

Step 3.2.2.1.2

Rewrite the expression.

$x + 4 = x \cdot x + 2 x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply $x$ by $x$ .

$x + 4 = x^{2} + 2 x$

Step 3.3

Solve the equation.

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Step 3.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x = x + 4$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 2 x - x = 4$

Step 3.3.2.2

Subtract $x$ from $2 x$ .

$x^{2} + x = 4$

Step 3.3.3

Subtract $4$ from both sides of the equation.

$x^{2} + x - 4 = 0$

Step 3.3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3.5

Substitute the values $a = 1$ , $b = 1$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 3.3.6

Simplify.

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.3.6.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

Step 3.3.6.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot 1}$

Step 3.3.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{17}}{2}$

Step 3.3.7

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{17}}{2}, - \frac{1 + \sqrt{17}}{2}$

Step 4

Exclude the solutions that do not make $\log (x + 4) - \log (x) = \log (x + 2)$ true.

$x = - \frac{1 - \sqrt{17}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{17}}{2}$

Decimal Form:

$x = 1.56155281 \dots$