QuestionQuestion 21, 4.4.87
HW Score: of 27 points
Points: 0 of 1
Solve the logarithmic equation. Be sure to reject any value of that is not in the domain of the original logarithmic expressions. Give the exact answer.
Solve the equation to find the solution set Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is .
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are infinitely many solutions.
C. There is no solution.
Studdy Solution
STEP 1
1. The equation involves logarithmic expressions.
2. The domain of the logarithmic functions requires that the arguments be positive.
3. We will use properties of logarithms to combine and solve the equation.
STEP 2
1. Use properties of logarithms to combine the logarithmic expressions.
2. Solve the resulting equation.
3. Check the solution(s) against the domain restrictions.
4. Determine the solution set.
STEP 3
Use the property of logarithms that states to combine the left side of the equation:
STEP 4
Since the logarithms are equal, their arguments must be equal. Set the arguments equal to each other:
STEP 5
Solve the quadratic equation:
Factor the quadratic equation:
Set each factor equal to zero and solve for :
STEP 6
Check the solutions against the domain restrictions. The original logarithmic expressions require and , which implies .
- For :
- is satisfied.
- is satisfied.
- For :
- is not satisfied.
Thus, is not in the domain and must be rejected.
STEP 7
Determine the solution set. The only valid solution is .
The solution set is:
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