Algebra Examples

$2 \log_{9} (x) = \log_{9} (8) + \log_{9} (x - 2)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify $2 \log_{9} (x)$ by moving $2$ inside the logarithm.

$\log_{9} (x^{2}) = \log_{9} (8) + \log_{9} (x - 2)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Simplify $\log_{9} (8) + \log_{9} (x - 2)$ .

Tap for more steps...

Step 2.1.1

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

$\log_{9} (x^{2}) = \log_{9} (8 (x - 2))$

Step 2.1.2

Apply the distributive property.

$\log_{9} (x^{2}) = \log_{9} (8 x + 8 \cdot - 2)$

Step 2.1.3

Multiply $8$ by $- 2$ .

$\log_{9} (x^{2}) = \log_{9} (8 x - 16)$

Step 3

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

$x^{2} = 8 x - 16$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Subtract $8 x$ from both sides of the equation.

$x^{2} - 8 x = - 16$

Step 4.2

Add $16$ to both sides of the equation.

$x^{2} - 8 x + 16 = 0$

Step 4.3

Factor using the perfect square rule.

Tap for more steps...

Step 4.3.1

Rewrite $16$ as $4^{2}$ .

$x^{2} - 8 x + 4^{2} = 0$

Step 4.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 4.3.3

Rewrite the polynomial.

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

Step 4.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

${(x - 4)}^{2} = 0$

Step 4.4

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.5

Add $4$ to both sides of the equation.

$x = 4$