Solve for x log base 4 of x- log base 4 of x-5 = log base 4 of 3

$\log_{4} (x) - \log_{4} (x - 5) = \log_{4} (3)$

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

$\log_{4} (\frac{x}{x - 5}) = \log_{4} (3)$

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

$\frac{x}{x - 5} = 3$

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.

$x - 5, 1$

Remove parentheses.

$x - 5, 1$

The LCM of one and any expression is the expression.

$x - 5$

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

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

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

Simplify the left side.

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Rewrite the expression.

$x = 3 (x - 5)$

Simplify the right side.

Apply the distributive property.

$x = 3 x + 3 \cdot - 5$

Multiply $3$ by $- 5$ .

$x = 3 x - 15$

Solve the equation.

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

Subtract $3 x$ from both sides of the equation.

$x - 3 x = - 15$

Subtract $3 x$ from $x$ .

$- 2 x = - 15$

Divide each term in $- 2 x = - 15$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 15$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 15}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 15}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 15}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{15}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{2}$

Decimal Form:

$x = 7.5$

Mixed Number Form:

$x = 7 \frac{1}{2}$

Solve for x log base 4 of x- log base 4 of x-5 = log base 4 of 3

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