Algebra Examples

$\log_{4} (x^{2}) - \log_{4} (x - 1) = 1$

Step 1

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

$\log_{4} (\frac{x^{2}}{x - 1}) = 1$

Step 2

Rewrite $\log_{4} (\frac{x^{2}}{x - 1}) = 1$ 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 $b^{y} = x$ .

$4^{1} = \frac{x^{2}}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$x^{2} = 4 (x - 1)$

Step 4

Simplify $4 (x - 1)$ .

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Step 4.1

Apply the distributive property.

$x^{2} = 4 x + 4 \cdot - 1$

Step 4.2

Multiply $4$ by $- 1$ .

$x^{2} = 4 x - 4$

Step 5

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

$x^{2} - 4 x = - 4$

Step 6

Factor $x$ out of $x^{2} - 4 x$ .

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Step 6.1

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

$x \cdot x - 4 x = - 4$

Step 6.2

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = - 4$

Step 6.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = - 4$

Step 7

Simplify $x (x - 4)$ .

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Step 7.1

Apply the distributive property.

$x \cdot x + x \cdot - 4 = - 4$

Step 7.2

Simplify the expression.

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Step 7.2.1

Multiply $x$ by $x$ .

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

Step 7.2.2

Move $- 4$ to the left of $x$ .

$x^{2} - 4 x = - 4$

Step 8

Add $4$ to both sides of the equation.

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

Step 9

Factor using the perfect square rule.

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Step 9.1

Rewrite $4$ as $2^{2}$ .

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

Step 9.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 9.3

Rewrite the polynomial.

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

Step 9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

${(x - 2)}^{2} = 0$

Step 10

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

$x - 2 = 0$

Step 11

Add $2$ to both sides of the equation.

$x = 2$