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Find the domain of $\log \left( {x - 5} \right)$ :
(A) $\left( {5,\infty } \right)$
(B) ${R^ + } - \left( 1 \right)$
(C) $R - \left( 1 \right)$
(D) None of these

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Last updated date: 18th Jun 2024
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Answer
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Hint: In this question we have to find the value $x$ where the function given in the question is defined. We all know that logarithm is defined only for the value of $x$ which is greater than $1$ . Therefore, we will put the variable part of the function given in the question greater than $1$ to find the domain of the function. The domain of the function is the value of $x$ where the function is defined.

Complete step-by-step solution:
The given function is $f\left( x \right) = \log \left( {x - 5} \right)$ .
For a function if we have to find the domain then find the value of $x$ where that particular function is defined
Now, to find the domain of the function we will put the variable part of the function greater than $1$ . Therefore, we can write:
$
  x - 5 > 0 \\
   \Rightarrow x > 5
 $
Now, from the above we can write $x \in \left( {5,\infty } \right)$ . Therefore, the domain of $\log \left( {x - 5} \right)$ is $\left( {5,\infty } \right)$ .

Hence, the correct option is (A).

Note: The domain of a logarithmic function is greater than $1$ and the range of the logarithmic function is the set of real numbers. For a function if we have to find the range then find the value of $y$ the function can take.
The important thing in this question is that we should have an idea about the value of $x$ where $\log $ is defined. Because if we don’t know that we will not be able to start the question. And be careful about whether to put an open bracket or close bracket on the values of $x$ because it will lead to incorrect answers.