Precalculus Examples

$\ln (- x + 1) - \ln (3 x + 5) = \ln (- 6 x + 1)$

Step 1

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

$\ln (\frac{- x + 1}{3 x + 5}) = \ln (- 6 x + 1)$

Step 2

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

$\frac{- x + 1}{3 x + 5} = - 6 x + 1$

Step 3

Solve for $x$ .

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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.

$3 x + 5, 1, 1$

Step 3.1.2

Remove parentheses.

$3 x + 5, 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$3 x + 5$

Step 3.2

Multiply each term in $\frac{- x + 1}{3 x + 5} = - 6 x + 1$ by $3 x + 5$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{- x + 1}{3 x + 5} = - 6 x + 1$ by $3 x + 5$ .

$\frac{- x + 1}{3 x + 5} (3 x + 5) = - 6 x (3 x + 5) + 1 (3 x + 5)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3 x + 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- x + 1}{3 x + 5} (3 x + 5) = - 6 x (3 x + 5) + 1 (3 x + 5)$

Step 3.2.2.1.2

Rewrite the expression.

$- x + 1 = - 6 x (3 x + 5) + 1 (3 x + 5)$

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.

$- x + 1 = - 6 x (3 x) - 6 x \cdot 5 + 1 (3 x + 5)$

Step 3.2.3.1.2

Rewrite using the commutative property of multiplication.

$- x + 1 = - 6 \cdot 3 x \cdot x - 6 x \cdot 5 + 1 (3 x + 5)$

Step 3.2.3.1.3

Multiply $5$ by $- 6$ .

$- x + 1 = - 6 \cdot 3 x \cdot x - 30 x + 1 (3 x + 5)$

Step 3.2.3.1.4

Simplify each term.

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Step 3.2.3.1.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.3.1.4.1.1

Move $x$ .

$- x + 1 = - 6 \cdot 3 (x \cdot x) - 30 x + 1 (3 x + 5)$

Step 3.2.3.1.4.1.2

Multiply $x$ by $x$ .

$- x + 1 = - 6 \cdot 3 x^{2} - 30 x + 1 (3 x + 5)$

Step 3.2.3.1.4.2

Multiply $- 6$ by $3$ .

$- x + 1 = - 18 x^{2} - 30 x + 1 (3 x + 5)$

Step 3.2.3.1.5

Multiply $3 x + 5$ by $1$ .

$- x + 1 = - 18 x^{2} - 30 x + 3 x + 5$

Step 3.2.3.2

Add $- 30 x$ and $3 x$ .

$- x + 1 = - 18 x^{2} - 27 x + 5$

Step 3.3

Solve the equation.

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Step 3.3.1

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

$- 18 x^{2} - 27 x + 5 = - x + 1$

Step 3.3.2

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

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Step 3.3.2.1

Add $x$ to both sides of the equation.

$- 18 x^{2} - 27 x + 5 + x = 1$

Step 3.3.2.2

Add $- 27 x$ and $x$ .

$- 18 x^{2} - 26 x + 5 = 1$

Step 3.3.3

Subtract $1$ from both sides of the equation.

$- 18 x^{2} - 26 x + 5 - 1 = 0$

Step 3.3.4

Subtract $1$ from $5$ .

$- 18 x^{2} - 26 x + 4 = 0$

Step 3.3.5

Factor $- 2$ out of $- 18 x^{2} - 26 x + 4$ .

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Step 3.3.5.1

Factor $- 2$ out of $- 18 x^{2}$ .

$- 2 (9 x^{2}) - 26 x + 4 = 0$

Step 3.3.5.2

Factor $- 2$ out of $- 26 x$ .

$- 2 (9 x^{2}) - 2 (13 x) + 4 = 0$

Step 3.3.5.3

Factor $- 2$ out of $4$ .

$- 2 (9 x^{2}) - 2 (13 x) - 2 \cdot - 2 = 0$

Step 3.3.5.4

Factor $- 2$ out of $- 2 (9 x^{2}) - 2 (13 x)$ .

$- 2 (9 x^{2} + 13 x) - 2 \cdot - 2 = 0$

Step 3.3.5.5

Factor $- 2$ out of $- 2 (9 x^{2} + 13 x) - 2 (- 2)$ .

$- 2 (9 x^{2} + 13 x - 2) = 0$

Step 3.3.6

Divide each term in $- 2 (9 x^{2} + 13 x - 2) = 0$ by $- 2$ and simplify.

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Step 3.3.6.1

Divide each term in $- 2 (9 x^{2} + 13 x - 2) = 0$ by $- 2$ .

$\frac{- 2 (9 x^{2} + 13 x - 2)}{- 2} = \frac{0}{- 2}$

Step 3.3.6.2

Simplify the left side.

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Step 3.3.6.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.6.2.1.1

Cancel the common factor.

$\frac{- 2 (9 x^{2} + 13 x - 2)}{- 2} = \frac{0}{- 2}$

Step 3.3.6.2.1.2

Divide $9 x^{2} + 13 x - 2$ by $1$ .

$9 x^{2} + 13 x - 2 = \frac{0}{- 2}$

Step 3.3.6.3

Simplify the right side.

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Step 3.3.6.3.1

Divide $0$ by $- 2$ .

$9 x^{2} + 13 x - 2 = 0$

Step 3.3.7

Use the quadratic formula to find the solutions.

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

Step 3.3.8

Substitute the values $a = 9$ , $b = 13$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (9 \cdot - 2)}}{2 \cdot 9}$

Step 3.3.9

Simplify.

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Step 3.3.9.1

Simplify the numerator.

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Step 3.3.9.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 9 \cdot - 2}}{2 \cdot 9}$

Step 3.3.9.1.2

Multiply $- 4 \cdot 9 \cdot - 2$ .

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Step 3.3.9.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 13 \pm \sqrt{169 - 36 \cdot - 2}}{2 \cdot 9}$

Step 3.3.9.1.2.2

Multiply $- 36$ by $- 2$ .

$x = \frac{- 13 \pm \sqrt{169 + 72}}{2 \cdot 9}$

Step 3.3.9.1.3

Add $169$ and $72$ .

$x = \frac{- 13 \pm \sqrt{241}}{2 \cdot 9}$

Step 3.3.9.2

Multiply $2$ by $9$ .

$x = \frac{- 13 \pm \sqrt{241}}{18}$

Step 3.3.10

The final answer is the combination of both solutions.

$x = - \frac{13 - \sqrt{241}}{18}, - \frac{13 + \sqrt{241}}{18}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{13 - \sqrt{241}}{18}, - \frac{13 + \sqrt{241}}{18}$

Decimal Form:

$x = 0.14023192 \dots, - 1.58467637 \dots$