Precalculus Examples

$\ln (- 2 y + 5) - \ln (y + 4) = \ln (- 11 y - 2)$

Step 1

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

$\ln (\frac{- 2 y + 5}{y + 4}) = \ln (- 11 y - 2)$

Step 2

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

$\frac{- 2 y + 5}{y + 4} = - 11 y - 2$

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

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

$y + 4, 1, 1$

Step 3.1.2

Remove parentheses.

$y + 4, 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$y + 4$

Step 3.2

Multiply each term in $\frac{- 2 y + 5}{y + 4} = - 11 y - 2$ by $y + 4$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{- 2 y + 5}{y + 4} = - 11 y - 2$ by $y + 4$ .

$\frac{- 2 y + 5}{y + 4} (y + 4) = - 11 y (y + 4) - 2 (y + 4)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y + 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 2 y + 5}{y + 4} (y + 4) = - 11 y (y + 4) - 2 (y + 4)$

Step 3.2.2.1.2

Rewrite the expression.

$- 2 y + 5 = - 11 y (y + 4) - 2 (y + 4)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Apply the distributive property.

$- 2 y + 5 = - 11 y \cdot y - 11 y \cdot 4 - 2 (y + 4)$

Step 3.2.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.3.1.2.1

Move $y$ .

$- 2 y + 5 = - 11 (y \cdot y) - 11 y \cdot 4 - 2 (y + 4)$

Step 3.2.3.1.2.2

Multiply $y$ by $y$ .

$- 2 y + 5 = - 11 y^{2} - 11 y \cdot 4 - 2 (y + 4)$

Step 3.2.3.1.3

Multiply $4$ by $- 11$ .

$- 2 y + 5 = - 11 y^{2} - 44 y - 2 (y + 4)$

Step 3.2.3.1.4

Apply the distributive property.

$- 2 y + 5 = - 11 y^{2} - 44 y - 2 y - 2 \cdot 4$

Step 3.2.3.1.5

Multiply $- 2$ by $4$ .

$- 2 y + 5 = - 11 y^{2} - 44 y - 2 y - 8$

Step 3.2.3.2

Subtract $2 y$ from $- 44 y$ .

$- 2 y + 5 = - 11 y^{2} - 46 y - 8$

Step 3.3

Solve the equation.

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Step 3.3.1

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

$- 11 y^{2} - 46 y - 8 = - 2 y + 5$

Step 3.3.2

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

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Step 3.3.2.1

Add $2 y$ to both sides of the equation.

$- 11 y^{2} - 46 y - 8 + 2 y = 5$

Step 3.3.2.2

Add $- 46 y$ and $2 y$ .

$- 11 y^{2} - 44 y - 8 = 5$

Step 3.3.3

Move all terms to the left side of the equation and simplify.

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Step 3.3.3.1

Subtract $5$ from both sides of the equation.

$- 11 y^{2} - 44 y - 8 - 5 = 0$

Step 3.3.3.2

Subtract $5$ from $- 8$ .

$- 11 y^{2} - 44 y - 13 = 0$

Step 3.3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3.5

Substitute the values $a = - 11$ , $b = - 44$ , and $c = - 13$ into the quadratic formula and solve for $y$ .

$\frac{44 \pm \sqrt{{(- 44)}^{2} - 4 \cdot (- 11 \cdot - 13)}}{2 \cdot - 11}$

Step 3.3.6

Simplify.

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Step 3.3.6.1

Simplify the numerator.

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Step 3.3.6.1.1

Raise $- 44$ to the power of $2$ .

$y = \frac{44 \pm \sqrt{1936 - 4 \cdot - 11 \cdot - 13}}{2 \cdot - 11}$

Step 3.3.6.1.2

Multiply $- 4 \cdot - 11 \cdot - 13$ .

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Step 3.3.6.1.2.1

Multiply $- 4$ by $- 11$ .

$y = \frac{44 \pm \sqrt{1936 + 44 \cdot - 13}}{2 \cdot - 11}$

Step 3.3.6.1.2.2

Multiply $44$ by $- 13$ .

$y = \frac{44 \pm \sqrt{1936 - 572}}{2 \cdot - 11}$

Step 3.3.6.1.3

Subtract $572$ from $1936$ .

$y = \frac{44 \pm \sqrt{1364}}{2 \cdot - 11}$

Step 3.3.6.1.4

Rewrite $1364$ as $2^{2} \cdot 341$ .

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Step 3.3.6.1.4.1

Factor $4$ out of $1364$ .

$y = \frac{44 \pm \sqrt{4 (341)}}{2 \cdot - 11}$

Step 3.3.6.1.4.2

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

$y = \frac{44 \pm \sqrt{2^{2} \cdot 341}}{2 \cdot - 11}$

Step 3.3.6.1.5

Pull terms out from under the radical.

$y = \frac{44 \pm 2 \sqrt{341}}{2 \cdot - 11}$

Step 3.3.6.2

Multiply $2$ by $- 11$ .

$y = \frac{44 \pm 2 \sqrt{341}}{- 22}$

Step 3.3.6.3

Simplify $\frac{44 \pm 2 \sqrt{341}}{- 22}$ .

$y = \frac{22 \pm \sqrt{341}}{- 11}$

Step 3.3.6.4

Move the negative in front of the fraction.

$y = - \frac{22 \pm \sqrt{341}}{11}$

Step 3.3.7

The final answer is the combination of both solutions.

$y = - \frac{22 + \sqrt{341}}{11}, - \frac{22 - \sqrt{341}}{11}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{22 + \sqrt{341}}{11}, - \frac{22 - \sqrt{341}}{11}$

Decimal Form:

$y = - 3.67874411 \dots, - 0.32125588 \dots$