For `y = log_(a)` x to be defined '`a`' must be

Options

A. any + ve real number
B. any number
C. `ge` e
D. any + ve real number `ne ` 1

Answer

To determine the conditions under which \( y = \log_a x \) is defined, we need to analyze the base \( a \) of the logarithm. The logarithm is defined under specific conditions regarding its base.

### Step-by-Step Solution:

1. **Understanding the Logarithm**:
The logarithm \( y = \log_a x \) is defined as the exponent to which the base \( a \) must be raised to produce the number \( x \). In mathematical terms, if \( y = \log_a x \), then \( a^y = x \).

2. **Condition for the Base \( a \)**:
For \( y = \log_a x \) to be defined, the base \( a \) must meet certain criteria:
- **Positive**: The base \( a \) must be a positive number. This is because logarithms with non-positive bases do not yield real numbers.
- **Not Equal to 1**: The base \( a \) cannot be equal to 1. If \( a = 1 \), then \( a^y = 1^y = 1 \) for any \( y \), which does not allow us to uniquely determine \( x \).

3. **Conclusion**:
Therefore, the base \( a \) must be a positive real number that is not equal to 1. This can be expressed mathematically as:
\[ a > 0 \quad \text{and} \quad a \neq 1 \]

### Final Answer:
The base \( a \) must be any positive number except for 1.