Question
Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer.
log Subscript 2 Baseline left parenthesis x minus 6 right parenthesis plus log Subscript 2 Baseline left parenthesis x minus 3 right parenthesis minus log Subscript 2 Baseline x equals 1
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Part 1
Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are infinitely many solutions.
C.
There is no solution.
log Subscript 2 Baseline left parenthesis x minus 6 right parenthesis plus log Subscript 2 Baseline left parenthesis x minus 3 right parenthesis minus log Subscript 2 Baseline x equals 1
Question content area bottom
Part 1
Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are infinitely many solutions.
C.
There is no solution.
Answers
GPT-4o mini
To solve the logarithmic equation \( \log_2 (x - 6) + \log_2 (x - 3) - \log_2 x = 1 \), we will first use the properties of logarithms to simplify the equation.
1. **Combine the logarithms:**
Using the property \( \log_a b + \log_a c = \log_a (bc) \), we can combine the first two logarithms:
\[
\log_2((x - 6)(x - 3)) - \log_2 x = 1
\]
Now using the property \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \):
\[
\log_2 \left( \frac{(x - 6)(x - 3)}{x} \right) = 1
\]
2. **Exponentiate both sides:**
This gives:
\[
\frac{(x - 6)(x - 3)}{x} = 2^1
\]
Simplifying this yields:
\[
\frac{(x - 6)(x - 3)}{x} = 2
\]
Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):
\[
(x - 6)(x - 3) = 2x
\]
3. **Expand and rearrange:**
Expanding the left side:
\[
x^2 - 3x - 6x + 18 = 2x
\]
Simplifying gives:
\[
x^2 - 9x + 18 = 2x
\]
Rearranging this leads to:
\[
x^2 - 11x + 18 = 0
\]
4. **Factor the quadratic:**
To factor \( x^2 - 11x + 18 \):
\[
(x - 2)(x - 9) = 0
\]
Thus, the solutions are:
\[
x = 2 \quad \text{or} \quad x = 9
\]
5. **Check the domain:**
The expressions \( \log_2(x - 6) \), \( \log_2(x - 3) \), and \( \log_2 x \) require that:
- \( x - 6 > 0 \) -> \( x > 6 \)
- \( x - 3 > 0 \) -> \( x > 3 \)
- \( x > 0 \)
Hence, \( x \) must be greater than 6. Thus, we reject \( x = 2 \) because it does not satisfy the domain requirements.
6. **Acceptable solution:**
The only acceptable solution is \( x = 9 \).
Thus, the final answer is:
The solution set is \( \{ 9 \} \).
1. **Combine the logarithms:**
Using the property \( \log_a b + \log_a c = \log_a (bc) \), we can combine the first two logarithms:
\[
\log_2((x - 6)(x - 3)) - \log_2 x = 1
\]
Now using the property \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \):
\[
\log_2 \left( \frac{(x - 6)(x - 3)}{x} \right) = 1
\]
2. **Exponentiate both sides:**
This gives:
\[
\frac{(x - 6)(x - 3)}{x} = 2^1
\]
Simplifying this yields:
\[
\frac{(x - 6)(x - 3)}{x} = 2
\]
Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):
\[
(x - 6)(x - 3) = 2x
\]
3. **Expand and rearrange:**
Expanding the left side:
\[
x^2 - 3x - 6x + 18 = 2x
\]
Simplifying gives:
\[
x^2 - 9x + 18 = 2x
\]
Rearranging this leads to:
\[
x^2 - 11x + 18 = 0
\]
4. **Factor the quadratic:**
To factor \( x^2 - 11x + 18 \):
\[
(x - 2)(x - 9) = 0
\]
Thus, the solutions are:
\[
x = 2 \quad \text{or} \quad x = 9
\]
5. **Check the domain:**
The expressions \( \log_2(x - 6) \), \( \log_2(x - 3) \), and \( \log_2 x \) require that:
- \( x - 6 > 0 \) -> \( x > 6 \)
- \( x - 3 > 0 \) -> \( x > 3 \)
- \( x > 0 \)
Hence, \( x \) must be greater than 6. Thus, we reject \( x = 2 \) because it does not satisfy the domain requirements.
6. **Acceptable solution:**
The only acceptable solution is \( x = 9 \).
Thus, the final answer is:
The solution set is \( \{ 9 \} \).