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If \[{\log _{\dfrac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0\], then find the exhaustive range of values of x.
A. \[\left( { - \infty ,{\rm{ }}2} \right) \cup \left( {3,{\rm{ }}\infty } \right)\]
B. \[\left( {2,3} \right)\]
C. \[\left( { - \infty ,{\rm{ }}1} \right) \cup \left( {1,{\rm{ }}2} \right) \cup \left( {2,{\rm{ }}\infty } \right)\]
D. None of these

Last updated date: 22nd Mar 2024
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MVSAT 2024
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Hint: In order to solve the question, first check whether the given term is positive. Next, write the condition for the given term to be positive. Finally, factorize and find the exhaustive range.

Formula used:
If \[\left( {x - a} \right)\left( {x - b} \right) < 0\], then \[x \in \left( {a,b} \right)\]

Complete step-by-step solution :
Given that
\[{\log _{\dfrac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0\]
If the base of the logarithm is less than 1 and the value of the logarithm is greater than 0, then the function must be less than 1.
That is the given value is positive. For the given value to be positive
 \[\left( {{x^2} - 5x + 7} \right) < 1\]
\[{x^2} - 5x + 6 < 0\]
\[\left( {x - 2} \right)\left( {x - 3} \right) < 0\]
That is
\[x \in \left( {2,3} \right)\]
Hence option B is the correct answer.

Additional information:
There are 2 types of logarithmic functions.
The logarithmic function with base 10 or 2 is known as a common logarithm.
The logarithmic function with base e is known as natural logarithmic.
The base of the logarithm can be a positive integer not equal to 1.
The argument of the logarithm is always greater than zero. A negative argument does not exist.

Note: Students can make mistakes while taking the condition for the given term to be positive. Remember that range means the set of all output values. Students should also be careful while factorizing the equation.