Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Multiply $x$ by $x$ .

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

Step 1.2.2.2

Move $- 1$ to the left of $x$ .

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

Step 1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} - x = 4^{\frac{1}{2}}$ .

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

Step 3.2

Simplify $4^{\frac{1}{2}}$ .

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

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

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

Step 3.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

$x^{2} - x = 2^{1}$

Step 3.2.3

Evaluate the exponent.

$x^{2} - x = 2$

Step 3.3

Subtract $2$ from both sides of the equation.

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

Step 3.4

Factor $x^{2} - x - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 3.4.2

Write the factored form using these integers.

$(x - 2) (x + 1) = 0$

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 $0$ .

$x - 2 = 0$

$x + 1 = 0$

Step 3.6

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

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Step 3.6.1

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

$x - 2 = 0$

Step 3.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.7

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.8

The final solution is all the values that make $(x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 4

Exclude the solutions that do not make $\log_{4} (x) + \log_{4} (x - 1) = \frac{1}{2}$ true.

$x = 2$