Algebra Examples

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

Step 1

Rewrite $\log_{x + 2} (9 x) = 2$ 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 $b^{y} = x$ .

${(x + 2)}^{2} = 9 x$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $9 x$ from both sides of the equation.

${(x + 2)}^{2} - 9 x = 0$

Step 2.1.2

Simplify each term.

Tap for more steps...

Step 2.1.2.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$(x + 2) (x + 2) - 9 x = 0$

Step 2.1.2.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.2.1

Apply the distributive property.

$x (x + 2) + 2 (x + 2) - 9 x = 0$

Step 2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 (x + 2) - 9 x = 0$

Step 2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - 9 x = 0$

Step 2.1.2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - 9 x = 0$

Step 2.1.2.3.1.2

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

$x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - 9 x = 0$

Step 2.1.2.3.1.3

Multiply $2$ by $2$ .

$x^{2} + 2 x + 2 x + 4 - 9 x = 0$

Step 2.1.2.3.2

Add $2 x$ and $2 x$ .

$x^{2} + 4 x + 4 - 9 x = 0$

Step 2.1.3

Subtract $9 x$ from $4 x$ .

$x^{2} - 5 x + 4 = 0$

Step 2.2

Factor $x^{2} - 5 x + 4$ using the AC method.

Tap for more steps...

Step 2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 2.2.2

Write the factored form using these integers.

$(x - 4) (x - 1) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x - 1 = 0$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.4.1

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

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.6

The final solution is all the values that make $(x - 4) (x - 1) = 0$ true.

$x = 4, 1$