Precalculus Examples

$\log (x) + \log (x + 1) = \log (x + 5)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (x (x + 1)) = \log (x + 5)$

Step 1.2

Apply the distributive property.

$\log (x \cdot x + x \cdot 1) = \log (x + 5)$

Step 1.3

Simplify the expression.

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Step 1.3.1

Multiply $x$ by $x$ .

$\log (x^{2} + x \cdot 1) = \log (x + 5)$

Step 1.3.2

Multiply $x$ by $1$ .

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

Step 2

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

$x^{2} + x = x + 5$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$x^{2} + x - x = 5$

Step 3.1.2

Combine the opposite terms in $x^{2} + x - x$ .

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Step 3.1.2.1

Subtract $x$ from $x$ .

$x^{2} + 0 = 5$

Step 3.1.2.2

Add $x^{2}$ and $0$ .

$x^{2} = 5$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{5}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Step 3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

Step 4

Exclude the solutions that do not make $\log (x) + \log (x + 1) = \log (x + 5)$ true.

$x = \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{5}$

Decimal Form:

$x = 2.23606797 \dots$