Precalculus Examples

$\ln (x - 8) - \ln (x + 7) = \ln (x - 10) - \ln (x + 8)$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Simplify the left side.

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Step 1.2.1

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

$\ln (\frac{x - 8}{x + 7}) = \ln (x - 10) - \ln (x + 8)$

Step 1.3

Simplify the right side.

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Step 1.3.1

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

$\ln (\frac{x - 8}{x + 7}) = \ln (\frac{x - 10}{x + 8})$

Step 1.4

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{x - 8}{x + 7} = \frac{x - 10}{x + 8}$

Step 1.5

Solve for $x$ .

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Step 1.5.1

Factor each term.

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Step 1.5.1.1

Split the fraction $\frac{x - 8}{x + 7}$ into two fractions.

$\frac{x}{x + 7} + \frac{- 8}{x + 7} = \frac{x - 10}{x + 8}$

Step 1.5.1.2

Split the fraction $\frac{x - 10}{x + 8}$ into two fractions.

$\frac{x}{x + 7} + \frac{- 8}{x + 7} = \frac{x}{x + 8} + \frac{- 10}{x + 8}$

Step 1.5.2

Find the LCD of the terms in the equation.

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Step 1.5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 7, x + 7, x + 8, x + 8$

Step 1.5.2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.5.2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.5.2.4

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.5.2.5

The factor for $x + 7$ is $x + 7$ itself.

$(x + 7) = x + 7$

$(x + 7)$ occurs $1$ time.

Step 1.5.2.6

The factor for $x + 8$ is $x + 8$ itself.

$(x + 8) = x + 8$

$(x + 8)$ occurs $1$ time.

Step 1.5.2.7

The LCM of $x + 7, x + 7, x + 8, x + 8$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 7) (x + 8)$

Step 1.5.3

Multiply each term in $\frac{x}{x + 7} + \frac{- 8}{x + 7} = \frac{x}{x + 8} + \frac{- 10}{x + 8}$ by $(x + 7) (x + 8)$ to eliminate the fractions.

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Step 1.5.3.1

Multiply each term in $\frac{x}{x + 7} + \frac{- 8}{x + 7} = \frac{x}{x + 8} + \frac{- 10}{x + 8}$ by $(x + 7) (x + 8)$ .

$\frac{x}{x + 7} ((x + 7) (x + 8)) + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Simplify each term.

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Step 1.5.3.2.1.1

Cancel the common factor of $x + 7$ .

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Step 1.5.3.2.1.1.1

Cancel the common factor.

$\frac{x}{x + 7} ((x + 7) (x + 8)) + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.1.2

Rewrite the expression.

$x (x + 8) + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.2

Apply the distributive property.

$x \cdot x + x \cdot 8 + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.3

Multiply $x$ by $x$ .

$x^{2} + x \cdot 8 + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.4

Move $8$ to the left of $x$ .

$x^{2} + 8 \cdot x + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.5

Cancel the common factor of $x + 7$ .

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Step 1.5.3.2.1.5.1

Cancel the common factor.

$x^{2} + 8 x + \frac{- 8}{x + 7} ((x + 7) (x + 8)) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.5.2

Rewrite the expression.

$x^{2} + 8 x - 8 (x + 8) = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.6

Apply the distributive property.

$x^{2} + 8 x - 8 x - 8 \cdot 8 = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.1.7

Multiply $- 8$ by $8$ .

$x^{2} + 8 x - 8 x - 64 = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.2

Combine the opposite terms in $x^{2} + 8 x - 8 x - 64$ .

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Step 1.5.3.2.2.1

Subtract $8 x$ from $8 x$ .

$x^{2} + 0 - 64 = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.2.2.2

Add $x^{2}$ and $0$ .

$x^{2} - 64 = \frac{x}{x + 8} ((x + 7) (x + 8)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Simplify each term.

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Step 1.5.3.3.1.1

Cancel the common factor of $x + 8$ .

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Step 1.5.3.3.1.1.1

Factor $x + 8$ out of $(x + 7) (x + 8)$ .

$x^{2} - 64 = \frac{x}{x + 8} ((x + 8) (x + 7)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.1.2

Cancel the common factor.

$x^{2} - 64 = \frac{x}{x + 8} ((x + 8) (x + 7)) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.1.3

Rewrite the expression.

$x^{2} - 64 = x (x + 7) + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.2

Apply the distributive property.

$x^{2} - 64 = x \cdot x + x \cdot 7 + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.3

Multiply $x$ by $x$ .

$x^{2} - 64 = x^{2} + x \cdot 7 + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.4

Move $7$ to the left of $x$ .

$x^{2} - 64 = x^{2} + 7 \cdot x + \frac{- 10}{x + 8} ((x + 7) (x + 8))$

Step 1.5.3.3.1.5

Cancel the common factor of $x + 8$ .

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Step 1.5.3.3.1.5.1

Factor $x + 8$ out of $(x + 7) (x + 8)$ .

$x^{2} - 64 = x^{2} + 7 x + \frac{- 10}{x + 8} ((x + 8) (x + 7))$

Step 1.5.3.3.1.5.2

Cancel the common factor.

$x^{2} - 64 = x^{2} + 7 x + \frac{- 10}{x + 8} ((x + 8) (x + 7))$

Step 1.5.3.3.1.5.3

Rewrite the expression.

$x^{2} - 64 = x^{2} + 7 x - 10 (x + 7)$

Step 1.5.3.3.1.6

Apply the distributive property.

$x^{2} - 64 = x^{2} + 7 x - 10 x - 10 \cdot 7$

Step 1.5.3.3.1.7

Multiply $- 10$ by $7$ .

$x^{2} - 64 = x^{2} + 7 x - 10 x - 70$

Step 1.5.3.3.2

Subtract $10 x$ from $7 x$ .

$x^{2} - 64 = x^{2} - 3 x - 70$

Step 1.5.4

Solve the equation.

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Step 1.5.4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 3 x - 70 = x^{2} - 64$

Step 1.5.4.2

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

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Step 1.5.4.2.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 3 x - 70 - x^{2} = - 64$

Step 1.5.4.2.2

Combine the opposite terms in $x^{2} - 3 x - 70 - x^{2}$ .

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Step 1.5.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- 3 x - 70 + 0 = - 64$

Step 1.5.4.2.2.2

Add $- 3 x - 70$ and $0$ .

$- 3 x - 70 = - 64$

Step 1.5.4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 1.5.4.3.1

Add $70$ to both sides of the equation.

$- 3 x = - 64 + 70$

Step 1.5.4.3.2

Add $- 64$ and $70$ .

$- 3 x = 6$

Step 1.5.4.4

Divide each term in $- 3 x = 6$ by $- 3$ and simplify.

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Step 1.5.4.4.1

Divide each term in $- 3 x = 6$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

Step 1.5.4.4.2

Simplify the left side.

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Step 1.5.4.4.2.1

Cancel the common factor of $- 3$ .

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Step 1.5.4.4.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

Step 1.5.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 3}$

Step 1.5.4.4.3

Simplify the right side.

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Step 1.5.4.4.3.1

Divide $6$ by $- 3$ .

$x = - 2$

Step 1.6

Exclude the solutions that do not make $\ln (\frac{x - 8}{x + 7}) = \ln (\frac{x - 10}{x + 8})$ true.

$No solution$

Step 2

The equation is not linear, so a constant slope does not exist.

Not Linear