Precalculus Examples

$2 \log_{2} (x) - \log_{2} (x + 5) = 4$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$2 \log_{2} (x) - \log_{2} (x + 5) = 4$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log_{2} (x) - \log_{2} (x + 5)$ .

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Step 2.1.1

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

$\log_{2} (x^{2}) - \log_{2} (x + 5) = 4$

Step 2.1.2

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

$\log_{2} (\frac{x^{2}}{x + 5}) = 4$

Step 3

Rewrite $\log_{2} (\frac{x^{2}}{x + 5}) = 4$ 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 $b^{y} = x$ .

$2^{4} = \frac{x^{2}}{x + 5}$

Step 4

Cross multiply to remove the fraction.

$x^{2} = 2^{4} (x + 5)$

Step 5

Simplify $2^{4} (x + 5)$ .

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Step 5.1

Raise $2$ to the power of $4$ .

$x^{2} = 16 (x + 5)$

Step 5.2

Apply the distributive property.

$x^{2} = 16 x + 16 \cdot 5$

Step 5.3

Multiply $16$ by $5$ .

$x^{2} = 16 x + 80$

Step 6

Subtract $16 x$ from both sides of the equation.

$x^{2} - 16 x = 80$

Step 7

Factor $x$ out of $x^{2} - 16 x$ .

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Step 7.1

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

$x \cdot x - 16 x = 80$

Step 7.2

Factor $x$ out of $- 16 x$ .

$x \cdot x + x \cdot - 16 = 80$

Step 7.3

Factor $x$ out of $x \cdot x + x \cdot - 16$ .

$x (x - 16) = 80$

Step 8

Simplify $x (x - 16)$ .

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Step 8.1

Apply the distributive property.

$x \cdot x + x \cdot - 16 = 80$

Step 8.2

Simplify the expression.

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Step 8.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 16 = 80$

Step 8.2.2

Move $- 16$ to the left of $x$ .

$x^{2} - 16 x = 80$

Step 9

Subtract $80$ from both sides of the equation.

$x^{2} - 16 x - 80 = 0$

Step 10

Factor $x^{2} - 16 x - 80$ using the AC method.

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Step 10.1

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 $- 80$ and whose sum is $- 16$ .

$- 20, 4$

Step 10.2

Write the factored form using these integers.

$(x - 20) (x + 4) = 0$

Step 11

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

$x - 20 = 0$

$x + 4 = 0$

Step 12

Set $x - 20$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Step 12.2

Add $20$ to both sides of the equation.

$x = 20$

Step 13

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

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Step 13.1

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

$x + 4 = 0$

Step 13.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 14

The final solution is all the values that make $(x - 20) (x + 4) = 0$ true.

$x = 20, - 4$

Step 15

Exclude the solutions that do not make $2 \log_{2} (x) - \log_{2} (x + 5) = 4$ true.

$x = 20$