Understanding the Domain of a Logarithmic Function

The question asks for the domain of the function defined by \(f(x) = \log_x 10\). The domain of a function is the set of all possible input values (in this case, values of \(x\)) for which the function is defined.

For a logarithmic function of the form \(\log_b a\), where \(b\) is the base and \(a\) is the argument, there are specific conditions that must be met for the function to be defined in the real number system. These conditions are:

The argument \(a\) must be positive (\(a > 0\)).

The base \(b\) must be positive (\(b > 0\)).

The base \(b\) cannot be equal to 1 (\(b \neq 1\)).

Applying Logarithm Domain Rules to \(f(x) = \log_x 10\)

Let's apply these conditions to the given function \(f(x) = \log_x 10\).

Here, the argument is \(a = 10\), and the base is \(b = x\).

Let's check each condition:

Argument condition: The argument must be positive. Here, the argument is 10. Since \(10 > 0\), this condition is already satisfied and does not impose any restriction on the value of \(x\).

Base condition (positivity): The base must be positive. Here, the base is \(x\). So, we must have \(x > 0\).

Base condition (not equal to 1): The base cannot be equal to 1. Here, the base is \(x\). So, we must have \(x \neq 1\).

For the function \(f(x) = \log_x 10\) to be defined, both conditions on the base must be satisfied simultaneously.

\(x > 0\)

\(x \neq 1\)

This means that \(x\) must be a positive real number, but it cannot be equal to 1.

Determining the Domain of \(f(x) = \log_x 10\)

Combining the conditions \(x > 0\) and \(x \neq 1\), the domain of the function \(f(x) = \log_x 10\) is the set of all positive real numbers except 1.

This can be written in set notation as \(\{x \in \mathbb{R} \mid x > 0 \text{ and } x \neq 1\}\).

Let's look at the given options:

Option 1: \(x > 10\). This is too restrictive. It doesn't include values between 0 and 10 (excluding 1).

Option 2: \(x > 0\) excluding \(x = 10\). This incorrectly excludes \(x=10\) (which is a valid input) and doesn't mention excluding \(x=1\).

Option 3: \(x \ge 10\). This is also too restrictive and doesn't account for the base requirements.

Option 4: \(x > 0\) excluding \(x = 1\). This exactly matches our derived domain conditions.

Therefore, the domain of the function \(f(x) = \log_x 10\) is \(x > 0\) excluding \(x = 1\).

Revision Table: Key Logarithm Properties

Additional Information on Function Domains

Understanding the domain of a function is crucial in mathematics. It tells us the set of valid inputs for which the function produces a real output. Different types of functions have different rules for determining their domain:

Polynomial Functions: The domain is typically all real numbers (\(\mathbb{R}\)).

Rational Functions: Functions of the form \(\frac{P(x)}{Q(x)}\) have a domain where the denominator \(Q(x)\) is not equal to zero.

Square Root Functions (or even roots): Functions of the form \(\sqrt{g(x)}\) have a domain where the expression inside the root, \(g(x)\), is non-negative (\(g(x) \ge 0\)).

Logarithmic Functions: As discussed, \(\log_b a\) requires \(a > 0\), \(b > 0\), and \(b \neq 1\).

When dealing with functions that are combinations of these types (e.g., a logarithm inside a square root), you must satisfy the domain conditions for all parts of the function simultaneously.