Calculus Examples

$\ln (x) - \ln (x - 1) = \ln (4 x - 6)$

Step 1

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

$\ln (\frac{x}{x - 1}) = \ln (4 x - 6)$

Step 2

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

$\frac{x}{x - 1} = 4 x - 6$

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.

$x - 1, 1, 1$

Step 3.1.2

Remove parentheses.

$x - 1, 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$x - 1$

Step 3.2

Multiply each term in $\frac{x}{x - 1} = 4 x - 6$ by $x - 1$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x}{x - 1} = 4 x - 6$ by $x - 1$ .

$\frac{x}{x - 1} (x - 1) = 4 x (x - 1) - 6 (x - 1)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x - 1$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{x}{x - 1} (x - 1) = 4 x (x - 1) - 6 (x - 1)$

Step 3.2.2.1.2

Rewrite the expression.

$x = 4 x (x - 1) - 6 (x - 1)$

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.

$x = 4 x \cdot x + 4 x \cdot - 1 - 6 (x - 1)$

Step 3.2.3.1.2

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

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Step 3.2.3.1.2.1

Move $x$ .

$x = 4 (x \cdot x) + 4 x \cdot - 1 - 6 (x - 1)$

Step 3.2.3.1.2.2

Multiply $x$ by $x$ .

$x = 4 x^{2} + 4 x \cdot - 1 - 6 (x - 1)$

Step 3.2.3.1.3

Multiply $- 1$ by $4$ .

$x = 4 x^{2} - 4 x - 6 (x - 1)$

Step 3.2.3.1.4

Apply the distributive property.

$x = 4 x^{2} - 4 x - 6 x - 6 \cdot - 1$

Step 3.2.3.1.5

Multiply $- 6$ by $- 1$ .

$x = 4 x^{2} - 4 x - 6 x + 6$

Step 3.2.3.2

Subtract $6 x$ from $- 4 x$ .

$x = 4 x^{2} - 10 x + 6$

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.

$4 x^{2} - 10 x + 6 = x$

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

Subtract $x$ from both sides of the equation.

$4 x^{2} - 10 x + 6 - x = 0$

Step 3.3.2.2

Subtract $x$ from $- 10 x$ .

$4 x^{2} - 11 x + 6 = 0$

Step 3.3.3

Factor by grouping.

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Step 3.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 6 = 24$ and whose sum is $b = - 11$ .

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Step 3.3.3.1.1

Factor $- 11$ out of $- 11 x$ .

$4 x^{2} - 11 x + 6 = 0$

Step 3.3.3.1.2

Rewrite $- 11$ as $- 3$ plus $- 8$

$4 x^{2} + (- 3 - 8) x + 6 = 0$

Step 3.3.3.1.3

Apply the distributive property.

$4 x^{2} - 3 x - 8 x + 6 = 0$

Step 3.3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.3.2.1

Group the first two terms and the last two terms.

$(4 x^{2} - 3 x) - 8 x + 6 = 0$

Step 3.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$x (4 x - 3) - 2 (4 x - 3) = 0$

Step 3.3.3.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

$(4 x - 3) (x - 2) = 0$

Step 3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x - 3 = 0$

$x - 2 = 0$

Step 3.3.5

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 3.3.5.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 3.3.5.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 3.3.5.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 3.3.5.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 3.3.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 3.3.6

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

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Step 3.3.6.1

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

$x - 2 = 0$

Step 3.3.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.3.7

The final solution is all the values that make $(4 x - 3) (x - 2) = 0$ true.

$x = \frac{3}{4}, 2$

Step 4

Exclude the solutions that do not make $\ln (x) - \ln (x - 1) = \ln (4 x - 6)$ true.

$x = 2$