Introduction:
To determine whether z is a purely real number when logeZ is real, we need to understand the properties of logarithms and the relationship between real and complex numbers.

Properties of logarithms:
1. The logarithm of a positive real number is always real.
2. The logarithm of a negative real number is undefined.

Understanding complex numbers:
A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√(-1)). The real part of a complex number is represented by a, and the imaginary part is represented by bi.

Analysis:
If logeZ is real, it means that the logarithm of Z to the base e is a real number. This implies that Z is a positive real number, as the logarithm of a negative real number is undefined.

If Z is a positive real number, it can be represented as a + 0i, where the imaginary part is zero. In this case, Z does not have an imaginary component and is purely real.

Conclusion:
Therefore, when logeZ is real, we can say that Z is a purely real number. This is because the logarithm of a positive real number is always real, and Z being a positive real number implies that it does not have an imaginary component.