Question

Explore the limitations of the values of b and x in equations of the form logbx=L
by determining which logarithm is undefined.

Answer 1

In equations of the form \( \log_b x = L \), we need to consider the constraints on the base \( b \) and the argument \( x \) to understand when the logarithm is defined.

### 1. Limitations on \( b \) (the base of the logarithm):
1. **Base must be positive**: \( b > 0 \)
- The logarithm is only defined for positive bases. A base of \( b \leq 0 \) would lead to undefined expressions when evaluating the logarithm.

2. **Base cannot be equal to 1**: \( b \neq 1 \)
- If \( b = 1 \), then the logarithmic function does not produce unique values, as \( \log_1 x \) would be undefined because it would not be possible to resolve \( 1^L = x \) for any \( x \).

### 2. Limitations on \( x \) (the argument of the logarithm):
1. **Argument must be positive**: \( x > 0 \)
- The logarithm is defined only for positive values of \( x \). If \( x \leq 0 \), the logarithm is undefined. This includes \( x = 0 \), which leads to negative infinity (\( \log_b 0 \) is undefined), and negative values of \( x \), which do not have real logarithmic values.

### Summary of Restrictions:
- The equation \( \log_b x = L \) is defined under the following conditions:
- The base \( b \) must be positive and not equal to 1: \( b > 0 \) and \( b \neq 1 \).
- The argument \( x \) must be positive: \( x > 0 \).

When these conditions are not met, the logarithmic expression \( \log_b x \) is considered undefined.