Precalculus Examples

$\ln (3 x - 4) = \ln (4) - \ln (x - 1)$

Step 1

Simplify the right side.

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Step 1.1

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

$\ln (3 x - 4) = \ln (\frac{4}{x - 1})$

Step 2

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

$3 x - 4 = \frac{4}{x - 1}$

Step 3

Solve for $x$ .

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Step 3.1

Add $4$ to both sides of the equation.

$3 x = \frac{4}{x - 1} + 4$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$1, x - 1, 1$

Step 3.2.2

Remove parentheses.

$1, x - 1, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$x - 1$

Step 3.3

Multiply each term in $3 x = \frac{4}{x - 1} + 4$ by $x - 1$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $3 x = \frac{4}{x - 1} + 4$ by $x - 1$ .

$3 x (x - 1) = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Apply the distributive property.

$3 x \cdot x + 3 x \cdot - 1 = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.2.2

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

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Step 3.3.2.2.1

Move $x$ .

$3 (x \cdot x) + 3 x \cdot - 1 = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.2.2.2

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot - 1 = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.2.3

Multiply $- 1$ by $3$ .

$3 x^{2} - 3 x = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $x - 1$ .

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Step 3.3.3.1.1.1

Cancel the common factor.

$3 x^{2} - 3 x = \frac{4}{x - 1} (x - 1) + 4 (x - 1)$

Step 3.3.3.1.1.2

Rewrite the expression.

$3 x^{2} - 3 x = 4 + 4 (x - 1)$

Step 3.3.3.1.2

Apply the distributive property.

$3 x^{2} - 3 x = 4 + 4 x + 4 \cdot - 1$

Step 3.3.3.1.3

Multiply $4$ by $- 1$ .

$3 x^{2} - 3 x = 4 + 4 x - 4$

Step 3.3.3.2

Combine the opposite terms in $4 + 4 x - 4$ .

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Step 3.3.3.2.1

Subtract $4$ from $4$ .

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

Step 3.3.3.2.2

Add $4 x$ and $0$ .

$3 x^{2} - 3 x = 4 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

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Step 3.4.1.1

Subtract $4 x$ from both sides of the equation.

$3 x^{2} - 3 x - 4 x = 0$

Step 3.4.1.2

Subtract $4 x$ from $- 3 x$ .

$3 x^{2} - 7 x = 0$

Step 3.4.2

Factor $x$ out of $3 x^{2} - 7 x$ .

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Step 3.4.2.1

Factor $x$ out of $3 x^{2}$ .

$x (3 x) - 7 x = 0$

Step 3.4.2.2

Factor $x$ out of $- 7 x$ .

$x (3 x) + x \cdot - 7 = 0$

Step 3.4.2.3

Factor $x$ out of $x (3 x) + x \cdot - 7$ .

$x (3 x - 7) = 0$

Step 3.4.3

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

$x = 0$

$3 x - 7 = 0$

Step 3.4.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.4.5

Set $3 x - 7$ equal to $0$ and solve for $x$ .

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Step 3.4.5.1

Set $3 x - 7$ equal to $0$ .

$3 x - 7 = 0$

Step 3.4.5.2

Solve $3 x - 7 = 0$ for $x$ .

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Step 3.4.5.2.1

Add $7$ to both sides of the equation.

$3 x = 7$

Step 3.4.5.2.2

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 3.4.5.2.2.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 3.4.5.2.2.2

Simplify the left side.

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Step 3.4.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 3.4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 3.4.6

The final solution is all the values that make $x (3 x - 7) = 0$ true.

$x = 0, \frac{7}{3}$

Step 4

Exclude the solutions that do not make $\ln (3 x - 4) = \ln (4) - \ln (x - 1)$ true.

$x = \frac{7}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{3}$

Decimal Form:

$x = 2.\overline{3}$

Mixed Number Form:

$x = 2 \frac{1}{3}$