Solve for x log of x-1- log of x-2 = log of 5

$\log (x - 1) - \log (x - 2) = \log (5)$

Step 1

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

$\log (\frac{x - 1}{x - 2}) = \log (5)$

Step 2

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

$\frac{x - 1}{x - 2} = 5$

Step 3

Solve for $x$ .

Find the LCD of the terms in the equation.

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

$x - 2, 1$

Remove parentheses.

$x - 2, 1$

The LCM of one and any expression is the expression.

$x - 2$

Multiply each term in $\frac{x - 1}{x - 2} = 5$ by $x - 2$ to eliminate the fractions.

Multiply each term in $\frac{x - 1}{x - 2} = 5$ by $x - 2$ .

$\frac{x - 1}{x - 2} (x - 2) = 5 (x - 2)$

Simplify the left side.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{x - 1}{x - 2} (x - 2) = 5 (x - 2)$

Rewrite the expression.

$x - 1 = 5 (x - 2)$

Simplify the right side.

Apply the distributive property.

$x - 1 = 5 x + 5 \cdot - 2$

Multiply $5$ by $- 2$ .

$x - 1 = 5 x - 10$

Solve the equation.

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

Subtract $5 x$ from both sides of the equation.

$x - 1 - 5 x = - 10$

Subtract $5 x$ from $x$ .

$- 4 x - 1 = - 10$

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

Add $1$ to both sides of the equation.

$- 4 x = - 10 + 1$

Add $- 10$ and $1$ .

$- 4 x = - 9$

Divide each term in $- 4 x = - 9$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 9$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 9}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 9}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 9}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{9}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{4}$

Decimal Form:

$x = 2.25$

Mixed Number Form:

$x = 2 \frac{1}{4}$

Solve for x log of x-1- log of x-2 = log of 5

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