Precalculus Examples

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

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{2} ((x - 10) (x - 3)) - \log_{2} (x) = 2$

Step 1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{(x - 10) (x - 3)}{x}) = 2$

Step 2

Rewrite $\log_{2} (\frac{(x - 10) (x - 3)}{x}) = 2$ 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 $b^{y} = x$ .

$2^{2} = \frac{(x - 10) (x - 3)}{x}$

Step 3

Cross multiply to remove the fraction.

$(x - 10) (x - 3) = 2^{2} (x)$

Step 4

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

$(x - 10) (x - 3) = 4 x$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $4 x$ from both sides of the equation.

$(x - 10) (x - 3) - 4 x = 0$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Expand $(x - 10) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 5.2.1.1

Apply the distributive property.

$x (x - 3) - 10 (x - 3) - 4 x = 0$

Step 5.2.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 10 (x - 3) - 4 x = 0$

Step 5.2.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 10 x - 10 \cdot - 3 - 4 x = 0$

Step 5.2.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 10 x - 10 \cdot - 3 - 4 x = 0$

Step 5.2.2.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 10 x - 10 \cdot - 3 - 4 x = 0$

Step 5.2.2.1.3

Multiply $- 10$ by $- 3$ .

$x^{2} - 3 x - 10 x + 30 - 4 x = 0$

Step 5.2.2.2

Subtract $10 x$ from $- 3 x$ .

$x^{2} - 13 x + 30 - 4 x = 0$

Step 5.3

Subtract $4 x$ from $- 13 x$ .

$x^{2} - 17 x + 30 = 0$

Step 6

Factor $x^{2} - 17 x + 30$ using the AC method.

Tap for more steps...

Step 6.1

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 $30$ and whose sum is $- 17$ .

$- 15, - 2$

Step 6.2

Write the factored form using these integers.

$(x - 15) (x - 2) = 0$

Step 7

Simplify $(x - 15) (x - 2)$ .

Tap for more steps...

Step 7.1

Expand $(x - 15) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 7.1.1

Apply the distributive property.

$x (x - 2) - 15 (x - 2) = 0$

Step 7.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 15 (x - 2) = 0$

Step 7.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 15 x - 15 \cdot - 2 = 0$

Step 7.2

Simplify and combine like terms.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 15 x - 15 \cdot - 2 = 0$

Step 7.2.1.2

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

$x^{2} - 2 \cdot x - 15 x - 15 \cdot - 2 = 0$

Step 7.2.1.3

Multiply $- 15$ by $- 2$ .

$x^{2} - 2 x - 15 x + 30 = 0$

Step 7.2.2

Subtract $15 x$ from $- 2 x$ .

$x^{2} - 17 x + 30 = 0$

Step 8

Factor $x^{2} - 17 x + 30$ using the AC method.

Tap for more steps...

Step 8.1

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 $30$ and whose sum is $- 17$ .

$- 15, - 2$

Step 8.2

Write the factored form using these integers.

$(x - 15) (x - 2) = 0$

Step 9

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

$x - 15 = 0$

$x - 2 = 0$

Step 10

Set $x - 15$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.1

Set $x - 15$ equal to $0$ .

$x - 15 = 0$

Step 10.2

Add $15$ to both sides of the equation.

$x = 15$

Step 11

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 11.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 11.2

Add $2$ to both sides of the equation.

$x = 2$

Step 12

The final solution is all the values that make $(x - 15) (x - 2) = 0$ true.

$x = 15, 2$

Step 13

Exclude the solutions that do not make $\log_{2} (x - 10) + \log_{2} (x - 3) - \log_{2} (x) = 2$ true.

$x = 15$