Solve for x log base 5 of x- log base 5 of x+4 = log base 5 of 38

$\log_{5} (x) - \log_{5} (x + 4) = \log_{5} (38)$

Step 1

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

$\log_{5} (\frac{x}{x + 4}) = \log_{5} (38)$

Step 2

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

$\frac{x}{x + 4} = 38$

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.

$x + 4, 1$

Remove parentheses.

$x + 4, 1$

The LCM of one and any expression is the expression.

$x + 4$

Multiply each term in $\frac{x}{x + 4} = 38$ by $x + 4$ to eliminate the fractions.

Multiply each term in $\frac{x}{x + 4} = 38$ by $x + 4$ .

$\frac{x}{x + 4} (x + 4) = 38 (x + 4)$

Simplify the left side.

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{x}{x + 4} (x + 4) = 38 (x + 4)$

Rewrite the expression.

$x = 38 (x + 4)$

Simplify the right side.

Apply the distributive property.

$x = 38 x + 38 \cdot 4$

Multiply $38$ by $4$ .

$x = 38 x + 152$

Solve the equation.

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

Subtract $38 x$ from both sides of the equation.

$x - 38 x = 152$

Subtract $38 x$ from $x$ .

$- 37 x = 152$

Divide each term in $- 37 x = 152$ by $- 37$ and simplify.

Divide each term in $- 37 x = 152$ by $- 37$ .

$\frac{- 37 x}{- 37} = \frac{152}{- 37}$

Simplify the left side.

Cancel the common factor of $- 37$ .

Cancel the common factor.

$\frac{- 37 x}{- 37} = \frac{152}{- 37}$

Divide $x$ by $1$ .

$x = \frac{152}{- 37}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{152}{37}$

Step 4

Exclude the solutions that do not make $\log_{5} (x) - \log_{5} (x + 4) = \log_{5} (38)$ true.

$No solution$

Solve for x log base 5 of x- log base 5 of x+4 = log base 5 of 38

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