Algebra Examples

$\log_{8} (w + 12) + \log_{8} (w) = 3 \cdot \log_{8} (4)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{8} ((w + 12) w) = 3 \cdot \log_{8} (4)$

Step 1.2

Apply the distributive property.

$\log_{8} (w \cdot w + 12 w) = 3 \cdot \log_{8} (4)$

Step 1.3

Multiply $w$ by $w$ .

$\log_{8} (w^{2} + 12 w) = 3 \cdot \log_{8} (4)$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Logarithm base $8$ of $4$ is $\frac{2}{3}$ .

$\log_{8} (w^{2} + 12 w) = 3 \cdot \frac{2}{3}$

Step 2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1

Cancel the common factor.

$\log_{8} (w^{2} + 12 w) = 3 \cdot \frac{2}{3}$

Step 2.2.2

Rewrite the expression.

$\log_{8} (w^{2} + 12 w) = 2$

Step 3

Rewrite $\log_{8} (w^{2} + 12 w) = 2$ 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 $b^{y} = x$ .

$8^{2} = w^{2} + 12 w$

Step 4

Solve for $w$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $w^{2} + 12 w = 8^{2}$ .

$w^{2} + 12 w = 8^{2}$

Step 4.2

Raise $8$ to the power of $2$ .

$w^{2} + 12 w = 64$

Step 4.3

Subtract $64$ from both sides of the equation.

$w^{2} + 12 w - 64 = 0$

Step 4.4

Factor $w^{2} + 12 w - 64$ using the AC method.

Tap for more steps...

Step 4.4.1

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 $- 64$ and whose sum is $12$ .

$- 4, 16$

Step 4.4.2

Write the factored form using these integers.

$(w - 4) (w + 16) = 0$

Step 4.5

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

$w - 4 = 0$

$w + 16 = 0$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$w - 4 = 0$

Step 4.6.2

Add $4$ to both sides of the equation.

$w = 4$

Step 4.7

Set $w + 16$ equal to $0$ and solve for $w$ .

Tap for more steps...

Step 4.7.1

Set $w + 16$ equal to $0$ .

$w + 16 = 0$

Step 4.7.2

Subtract $16$ from both sides of the equation.

$w = - 16$

Step 4.8

The final solution is all the values that make $(w - 4) (w + 16) = 0$ true.

$w = 4, - 16$

Step 5

Exclude the solutions that do not make $\log_{8} (w + 12) + \log_{8} (w) = 3 \cdot \log_{8} (4)$ true.

$w = 4$