Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

The functions \(f(x)=\log (x-1)-\log (x-2)\) and \(g(x)=\) \(\log \left(\frac{x-1}{x-2}\right)\) are identical when \(x\) lies in the interval (A) \([1,2]\) (B) \([2, \infty]\) (C) \((2, \infty)\) (D) \((-\infty, \infty)\)

Step-by-Step Solution

Answer

(C) (2, ∞)

Step 1: Understand the Function f(x)

Start by evaluating the function \( f(x) = \log (x-1) - \log (x-2) \). This can be simplified using the logarithm property \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), leading to \( f(x) = \log\left(\frac{x-1}{x-2}\right) \).

Step 2: Compare with g(x)

The function \( g(x) = \log \left( \frac{x-1}{x-2} \right) \) is given and is identical to the simplified form of \( f(x) \). Therefore, \( f(x) = g(x) \).

Step 3: Determine the Domain of f(x) and g(x)

For \( f(x) \) and \( g(x) \) to be defined, the expressions inside the logarithms, \( x-1 \) and \( x-2 \), must be positive. This requires \( x-1 > 0 \rightarrow x > 1 \) and \( x-2 > 0 \rightarrow x > 2 \). Hence, the domain of these functions is \( x > 2 \).

Step 4: Choose the Correct Interval

Based on the domain determined, these functions are defined when \( x \) lies in the interval \((2, \infty)\). This corresponds to option (C).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions involving logarithms, which are the inverse operations of exponential functions. A logarithm answers the question: "To what exponent must we raise a specific base to obtain a given number?" In mathematical terms, if we have a logarithm of the form \[ \log_b(a) = c \]where \( b \) is the base, \( a \) is the argument, and \( c \) is the exponent. This means that \( b^c = a \).

• The base of a logarithm must be a positive number, not equal to one.
• The argument (inside the logarithm) must be positive.
• Common bases are \( e \) (natural logarithm) and 10 (common logarithm).

Logarithmic functions follow specific rules, such as the product, quotient, and power rules. In the exercise, the quotient rule for logarithms is applied, which states that \( \log(a) - \log(b) = \log \left(\frac{a}{b}\right) \). This rule simplifies expressions by converting them into single logarithmic statements, facilitating easier comparison and evaluation.

Function Domain

The domain of a function is the set of all possible input values (usually \( x \)) for which the function is defined. For logarithmic functions, the domain is determined by the condition that the argument inside the logarithm must be greater than zero because a logarithm of zero or a negative number is undefined in real numbers.
In the context of the exercise, to find the domain of the given functions \( f(x) = \log(x-1) - \log(x-2) \) or equivalently \( g(x) = \log \left(\frac{x-1}{x-2}\right) \), the expressions \( x-1 \) and \( x-2 \) must be positive:

• For \( x-1 > 0 \): \( x > 1 \)
• For \( x-2 > 0 \): \( x > 2 \)

The more restrictive condition "\( x > 2 \)" becomes the domain, indicating that both functions are only defined when \( x \) is greater than 2. This careful consideration of the function domain ensures that the logarithmic expressions remain valid and that mathematical operations can be accurately performed.

Interval Notation

Interval notation is a mathematical shorthand used to describe continuous sets of real numbers, often representing the domain or range of a function. It employs brackets and parentheses to specify which endpoint values are included or excluded.

• "[ ]" brackets signify that the endpoint is included (closed interval).
• "( )" parentheses suggest that the endpoint is not included (open interval).

For the exercise, the domain of the functions \( f(x) \) and \( g(x) \) is expressed as \( (2, \infty) \) in interval notation. This denotes all real numbers greater than 2, but not including 2 itself. Therefore, parentheses are used around 2. It essential for students to understand how interval notation succinctly captures the range of input values for which a function is defined, providing clear and concise information about a function's domain.