Algebra Examples

$2 \log (2 x + 2) = \log (16) + 2 \log (x - 2)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify $2 \log (2 x + 2)$ by moving $2$ inside the logarithm.

$\log ({(2 x + 2)}^{2}) = \log (16) + 2 \log (x - 2)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Simplify $\log (16) + 2 \log (x - 2)$ .

Tap for more steps...

Step 2.1.1

Simplify $2 \log (x - 2)$ by moving $2$ inside the logarithm.

$\log ({(2 x + 2)}^{2}) = \log (16) + \log ({(x - 2)}^{2})$

Step 2.1.2

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

$\log ({(2 x + 2)}^{2}) = \log (16 {(x - 2)}^{2})$

Step 3

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

${(2 x + 2)}^{2} = 16 {(x - 2)}^{2}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Simplify $16 {(x - 2)}^{2}$ .

Tap for more steps...

Step 4.1.1

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

${(2 x + 2)}^{2} = 16 ((x - 2) (x - 2))$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Apply the distributive property.

${(2 x + 2)}^{2} = 16 (x (x - 2) - 2 (x - 2))$

Step 4.1.2.2

Apply the distributive property.

${(2 x + 2)}^{2} = 16 (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Step 4.1.2.3

Apply the distributive property.

${(2 x + 2)}^{2} = 16 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 4.1.3

Simplify and combine like terms.

Tap for more steps...

Step 4.1.3.1

Simplify each term.

Tap for more steps...

Step 4.1.3.1.1

Multiply $x$ by $x$ .

${(2 x + 2)}^{2} = 16 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 4.1.3.1.2

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

${(2 x + 2)}^{2} = 16 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Step 4.1.3.1.3

Multiply $- 2$ by $- 2$ .

${(2 x + 2)}^{2} = 16 (x^{2} - 2 x - 2 x + 4)$

Step 4.1.3.2

Subtract $2 x$ from $- 2 x$ .

${(2 x + 2)}^{2} = 16 (x^{2} - 4 x + 4)$

Step 4.1.4

Apply the distributive property.

${(2 x + 2)}^{2} = 16 x^{2} + 16 (- 4 x) + 16 \cdot 4$

Step 4.1.5

Simplify.

Tap for more steps...

Step 4.1.5.1

Multiply $- 4$ by $16$ .

${(2 x + 2)}^{2} = 16 x^{2} - 64 x + 16 \cdot 4$

Step 4.1.5.2

Multiply $16$ by $4$ .

${(2 x + 2)}^{2} = 16 x^{2} - 64 x + 64$

Step 4.2

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

$16 x^{2} - 64 x + 64 = {(2 x + 2)}^{2}$

Step 4.3

Simplify ${(2 x + 2)}^{2}$ .

Tap for more steps...

Step 4.3.1

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

$16 x^{2} - 64 x + 64 = (2 x + 2) (2 x + 2)$

Step 4.3.2

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

Tap for more steps...

Step 4.3.2.1

Apply the distributive property.

$16 x^{2} - 64 x + 64 = 2 x (2 x + 2) + 2 (2 x + 2)$

Step 4.3.2.2

Apply the distributive property.

$16 x^{2} - 64 x + 64 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x + 2)$

Step 4.3.2.3

Apply the distributive property.

$16 x^{2} - 64 x + 64 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Step 4.3.3

Simplify and combine like terms.

Tap for more steps...

Step 4.3.3.1

Simplify each term.

Tap for more steps...

Step 4.3.3.1.1

Rewrite using the commutative property of multiplication.

$16 x^{2} - 64 x + 64 = 2 \cdot 2 x \cdot x + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Step 4.3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.3.3.1.2.1

Move $x$ .

$16 x^{2} - 64 x + 64 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Step 4.3.3.1.2.2

Multiply $x$ by $x$ .

$16 x^{2} - 64 x + 64 = 2 \cdot 2 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Step 4.3.3.1.3

Multiply $2$ by $2$ .

$16 x^{2} - 64 x + 64 = 4 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Step 4.3.3.1.4

Multiply $2$ by $2$ .

$16 x^{2} - 64 x + 64 = 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot 2$

Step 4.3.3.1.5

Multiply $2$ by $2$ .

$16 x^{2} - 64 x + 64 = 4 x^{2} + 4 x + 4 x + 2 \cdot 2$

Step 4.3.3.1.6

Multiply $2$ by $2$ .

$16 x^{2} - 64 x + 64 = 4 x^{2} + 4 x + 4 x + 4$

Step 4.3.3.2

Add $4 x$ and $4 x$ .

$16 x^{2} - 64 x + 64 = 4 x^{2} + 8 x + 4$

Step 4.4

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

Tap for more steps...

Step 4.4.1

Subtract $4 x^{2}$ from both sides of the equation.

$16 x^{2} - 64 x + 64 - 4 x^{2} = 8 x + 4$

Step 4.4.2

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

$16 x^{2} - 64 x + 64 - 4 x^{2} - 8 x = 4$

Step 4.4.3

Subtract $4 x^{2}$ from $16 x^{2}$ .

$12 x^{2} - 64 x + 64 - 8 x = 4$

Step 4.4.4

Subtract $8 x$ from $- 64 x$ .

$12 x^{2} - 72 x + 64 = 4$

Step 4.5

Subtract $4$ from both sides of the equation.

$12 x^{2} - 72 x + 64 - 4 = 0$

Step 4.6

Subtract $4$ from $64$ .

$12 x^{2} - 72 x + 60 = 0$

Step 4.7

Factor the left side of the equation.

Tap for more steps...

Step 4.7.1

Factor $12$ out of $12 x^{2} - 72 x + 60$ .

Tap for more steps...

Step 4.7.1.1

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) - 72 x + 60 = 0$

Step 4.7.1.2

Factor $12$ out of $- 72 x$ .

$12 (x^{2}) + 12 (- 6 x) + 60 = 0$

Step 4.7.1.3

Factor $12$ out of $60$ .

$12 x^{2} + 12 (- 6 x) + 12 \cdot 5 = 0$

Step 4.7.1.4

Factor $12$ out of $12 x^{2} + 12 (- 6 x)$ .

$12 (x^{2} - 6 x) + 12 \cdot 5 = 0$

Step 4.7.1.5

Factor $12$ out of $12 (x^{2} - 6 x) + 12 \cdot 5$ .

$12 (x^{2} - 6 x + 5) = 0$

Step 4.7.2

Factor.

Tap for more steps...

Step 4.7.2.1

Factor $x^{2} - 6 x + 5$ using the AC method.

Tap for more steps...

Step 4.7.2.1.1

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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 4.7.2.1.2

Write the factored form using these integers.

$12 ((x - 5) (x - 1)) = 0$

Step 4.7.2.2

Remove unnecessary parentheses.

$12 (x - 5) (x - 1) = 0$

Step 4.8

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

$x - 5 = 0$

$x - 1 = 0$

Step 4.9

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.9.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.9.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.10

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

Tap for more steps...

Step 4.10.1

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

$x - 1 = 0$

Step 4.10.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.11

The final solution is all the values that make $12 (x - 5) (x - 1) = 0$ true.

$x = 5, 1$

Step 5

Exclude the solutions that do not make $2 \log (2 x + 2) = \log (16) + 2 \log (x - 2)$ true.

$x = 5$