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To solve the equation
[tex]\[ \log_2(x - \sqrt{2}) + \log_2(x + \sqrt{2}) - \log_2(x - 1) = 1, \][/tex]
we can proceed with the following steps:
1. Combine the logarithmic terms using the properties of logarithms:
[tex]\[ \log_2(x - \sqrt{2}) + \log_2(x + \sqrt{2}) = \log_2\big((x - \sqrt{2})(x + \sqrt{2})\big). \][/tex]
The product inside the log becomes:
[tex]\[ (x - \sqrt{2})(x + \sqrt{2}) = x^2 - (\sqrt{2})^2 = x^2 - 2. \][/tex]
Thus, we can rewrite the left-hand side of the equation as:
[tex]\[ \log_2(x^2 - 2) - \log_2(x - 1). \][/tex]
2. Apply the property of logarithms that states [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex]:
[tex]\[ \log_2\left(\frac{x^2 - 2}{x - 1}\right). \][/tex]
The equation now becomes:
[tex]\[ \log_2\left(\frac{x^2 - 2}{x - 1}\right) = 1. \][/tex]
3. Rewrite the logarithmic equation in exponential form to remove the logarithm:
[tex]\[ \frac{x^2 - 2}{x - 1} = 2^1. \][/tex]
Simplify this to:
[tex]\[ \frac{x^2 - 2}{x - 1} = 2. \][/tex]
4. Solve the resulting rational equation:
[tex]\[ x^2 - 2 = 2(x - 1). \][/tex]
Distribute and rearrange the terms:
[tex]\[ x^2 - 2 = 2x - 2. \][/tex]
[tex]\[ x^2 - 2x = 0. \][/tex]
[tex]\[ x(x - 2) = 0. \][/tex]
5. Find the solutions by solving for [tex]\(x\)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad x = 2. \][/tex]
6. Check for extraneous solutions by substituting the solutions back into the original equation to ensure they do not violate any constraints (i.e., the arguments of the logarithms must be positive):
- For [tex]\(x = 0\)[/tex]:
[tex]\[ \log_2(0 - \sqrt{2}) \quad (\text{undefined, because } 0 - \sqrt{2} \text{ is negative}). \][/tex]
- Hence, discard [tex]\(x = 0\)[/tex] as it is not valid.
- For [tex]\(x = 2\)[/tex]:
Check the original equation:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) - \log_2(2 - 1). \][/tex]
This simplifies to:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) - \log_2(1). \][/tex]
Since [tex]\(\log_2(1) = 0\)[/tex], we get:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) = 1. \][/tex]
Let’s check this calculation and ensure it satisfies the given equation.
7. Verify [tex]\(x = 2\)[/tex] satisfies the equation:
- [tex]\((2 - \sqrt{2}) > 0\)[/tex], [tex]\((2 + \sqrt{2}) > 0\)[/tex], [tex]\((2 - 1) = 1\)[/tex] (all are positive).
- Thus, [tex]\(\boxed{2}\)[/tex] is indeed a valid solution.
Therefore, the solution to the equation is [tex]\( \boxed{2} \)[/tex].
[tex]\[ \log_2(x - \sqrt{2}) + \log_2(x + \sqrt{2}) - \log_2(x - 1) = 1, \][/tex]
we can proceed with the following steps:
1. Combine the logarithmic terms using the properties of logarithms:
[tex]\[ \log_2(x - \sqrt{2}) + \log_2(x + \sqrt{2}) = \log_2\big((x - \sqrt{2})(x + \sqrt{2})\big). \][/tex]
The product inside the log becomes:
[tex]\[ (x - \sqrt{2})(x + \sqrt{2}) = x^2 - (\sqrt{2})^2 = x^2 - 2. \][/tex]
Thus, we can rewrite the left-hand side of the equation as:
[tex]\[ \log_2(x^2 - 2) - \log_2(x - 1). \][/tex]
2. Apply the property of logarithms that states [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex]:
[tex]\[ \log_2\left(\frac{x^2 - 2}{x - 1}\right). \][/tex]
The equation now becomes:
[tex]\[ \log_2\left(\frac{x^2 - 2}{x - 1}\right) = 1. \][/tex]
3. Rewrite the logarithmic equation in exponential form to remove the logarithm:
[tex]\[ \frac{x^2 - 2}{x - 1} = 2^1. \][/tex]
Simplify this to:
[tex]\[ \frac{x^2 - 2}{x - 1} = 2. \][/tex]
4. Solve the resulting rational equation:
[tex]\[ x^2 - 2 = 2(x - 1). \][/tex]
Distribute and rearrange the terms:
[tex]\[ x^2 - 2 = 2x - 2. \][/tex]
[tex]\[ x^2 - 2x = 0. \][/tex]
[tex]\[ x(x - 2) = 0. \][/tex]
5. Find the solutions by solving for [tex]\(x\)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad x = 2. \][/tex]
6. Check for extraneous solutions by substituting the solutions back into the original equation to ensure they do not violate any constraints (i.e., the arguments of the logarithms must be positive):
- For [tex]\(x = 0\)[/tex]:
[tex]\[ \log_2(0 - \sqrt{2}) \quad (\text{undefined, because } 0 - \sqrt{2} \text{ is negative}). \][/tex]
- Hence, discard [tex]\(x = 0\)[/tex] as it is not valid.
- For [tex]\(x = 2\)[/tex]:
Check the original equation:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) - \log_2(2 - 1). \][/tex]
This simplifies to:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) - \log_2(1). \][/tex]
Since [tex]\(\log_2(1) = 0\)[/tex], we get:
[tex]\[ \log_2(2 - \sqrt{2}) + \log_2(2 + \sqrt{2}) = 1. \][/tex]
Let’s check this calculation and ensure it satisfies the given equation.
7. Verify [tex]\(x = 2\)[/tex] satisfies the equation:
- [tex]\((2 - \sqrt{2}) > 0\)[/tex], [tex]\((2 + \sqrt{2}) > 0\)[/tex], [tex]\((2 - 1) = 1\)[/tex] (all are positive).
- Thus, [tex]\(\boxed{2}\)[/tex] is indeed a valid solution.
Therefore, the solution to the equation is [tex]\( \boxed{2} \)[/tex].
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