Finite Math Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{2} (r (r - 2)) = 3$

Step 1.2

Apply the distributive property.

$\log_{2} (r \cdot r + r \cdot - 2) = 3$

Step 1.3

Simplify the expression.

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Step 1.3.1

Multiply $r$ by $r$ .

$\log_{2} (r^{2} + r \cdot - 2) = 3$

Step 1.3.2

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

$\log_{2} (r^{2} - 2 r) = 3$

Step 2

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

$2^{3} = r^{2} - 2 r$

Step 3

Solve for $r$ .

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Step 3.1

Rewrite the equation as $r^{2} - 2 r = 2^{3}$ .

$r^{2} - 2 r = 2^{3}$

Step 3.2

Raise $2$ to the power of $3$ .

$r^{2} - 2 r = 8$

Step 3.3

Subtract $8$ from both sides of the equation.

$r^{2} - 2 r - 8 = 0$

Step 3.4

Factor $r^{2} - 2 r - 8$ using the AC method.

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Step 3.4.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.4.2

Write the factored form using these integers.

$(r - 4) (r + 2) = 0$

Step 3.5

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

$r - 4 = 0$

$r + 2 = 0$

Step 3.6

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

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Step 3.6.1

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

$r - 4 = 0$

Step 3.6.2

Add $4$ to both sides of the equation.

$r = 4$

Step 3.7

Set $r + 2$ equal to $0$ and solve for $r$ .

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Step 3.7.1

Set $r + 2$ equal to $0$ .

$r + 2 = 0$

Step 3.7.2

Subtract $2$ from both sides of the equation.

$r = - 2$

Step 3.8

The final solution is all the values that make $(r - 4) (r + 2) = 0$ true.

$r = 4, - 2$

Step 4

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

$r = 4$