Algebra Examples

$\log_{5} (x + 10) + \log_{5} (x - 10) = 3$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{5} ((x + 10) (x - 10)) = 3$

Step 1.2

Expand $(x + 10) (x - 10)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\log_{5} (x (x - 10) + 10 (x - 10)) = 3$

Step 1.2.2

Apply the distributive property.

$\log_{5} (x \cdot x + x \cdot - 10 + 10 (x - 10)) = 3$

Step 1.2.3

Apply the distributive property.

$\log_{5} (x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10) = 3$

Step 1.3

Simplify terms.

Tap for more steps...

Step 1.3.1

Combine the opposite terms in $x \cdot x + x \cdot - 10 + 10 x + 10 \cdot - 10$ .

Tap for more steps...

Step 1.3.1.1

Reorder the factors in the terms $x \cdot - 10$ and $10 x$ .

$\log_{5} (x \cdot x - 10 x + 10 x + 10 \cdot - 10) = 3$

Step 1.3.1.2

Add $- 10 x$ and $10 x$ .

$\log_{5} (x \cdot x + 0 + 10 \cdot - 10) = 3$

Step 1.3.1.3

Add $x \cdot x$ and $0$ .

$\log_{5} (x \cdot x + 10 \cdot - 10) = 3$

Step 1.3.2

Simplify each term.

Tap for more steps...

Step 1.3.2.1

Multiply $x$ by $x$ .

$\log_{5} (x^{2} + 10 \cdot - 10) = 3$

Step 1.3.2.2

Multiply $10$ by $- 10$ .

$\log_{5} (x^{2} - 100) = 3$

Step 2

Rewrite $\log_{5} (x^{2} - 100) = 3$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$5^{3} = x^{2} - 100$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $x^{2} - 100 = 5^{3}$ .

$x^{2} - 100 = 5^{3}$

Step 3.2

Raise $5$ to the power of $3$ .

$x^{2} - 100 = 125$

Step 3.3

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

Tap for more steps...

Step 3.3.1

Add $100$ to both sides of the equation.

$x^{2} = 125 + 100$

Step 3.3.2

Add $125$ and $100$ .

$x^{2} = 225$

Step 3.4

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

$x = \pm \sqrt{225}$

Step 3.5

Simplify $\pm \sqrt{225}$ .

Tap for more steps...

Step 3.5.1

Rewrite $225$ as $15^{2}$ .

$x = \pm \sqrt{15^{2}}$

Step 3.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 15$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$x = 15$

Step 3.6.2

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

$x = - 15$

Step 3.6.3

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

$x = 15, - 15$

Step 4

Exclude the solutions that do not make $\log_{5} (x + 10) + \log_{5} (x - 10) = 3$ true.

$x = 15$