QuestionSolve for $x$ . $\log _{3}(2 x-5)-\log _{3}(x+1)=2$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ 3 . (Simplify your answer. Type an integer or a fraction. Use a comma $\qquad$ para. vers as needed.) B. The solution set is the empty set.

Studdy Solution

STEP 1

1. The equation involves logarithms with the same base, which allows us to use logarithmic properties.
2. The domain of the logarithmic functions requires that the arguments be positive: $2x - 5 > 0$ and $x + 1 > 0$ .

STEP 2

1. Use logarithmic properties to combine the logarithms.
2. Solve the resulting equation for $x$ .
3. Check the solution against the domain restrictions.

STEP 3

Use the property of logarithms that states $\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$ to combine the logarithms:
$\log_{3}(2x - 5) - \log_{3}(x + 1) = \log_{3}\left(\frac{2x - 5}{x + 1}\right)$
This simplifies the equation to:
$\log_{3}\left(\frac{2x - 5}{x + 1}\right) = 2$

STEP 4

To eliminate the logarithm, rewrite the equation in exponential form. If $\log_b(A) = C$ , then $A = b^C$ . Thus:
$\frac{2x - 5}{x + 1} = 3^2$
Simplify the right side:
$\frac{2x - 5}{x + 1} = 9$

STEP 5

Solve for $x$ by clearing the fraction. Multiply both sides by $x + 1$ :
$2x - 5 = 9(x + 1)$
Expand the right side:
$2x - 5 = 9x + 9$
Rearrange the equation to isolate terms involving $x$ :
$2x - 9x = 9 + 5$
$-7x = 14$
Divide both sides by $-7$ :
$x = -2$

STEP 6

Check the solution against the domain restrictions. The original expressions $2x - 5$ and $x + 1$ must be positive:
1. $2x - 5 > 0$ implies $x > \frac{5}{2}$ .
2. $x + 1 > 0$ implies $x > -1$ .
Since $x = -2$ does not satisfy these domain conditions, it is not a valid solution.
The solution set is the empty set.
B. The solution set is the empty set.

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