Coefficient of variation is defined as the ratio of standard deviation to the arithmetic mean. Coefficient of variation gives a sense of "relative variability," as reported by the GraphPad Statistical software website. It can be expressed either as a fraction or a percent.
Coefficient of variance (CV) is used to understand the scatter of variables that are expressed in different units. For example, the coefficient of variation for blood pressure can be compared with the coefficient of variation for pulse rate. In this case, blood pressure and pulse rate are two different variables.
While interpreting coefficient of variation, 0 can be reported provided it actually implies "zero." For example, zero weight implies no weight. Coefficient of variation can be reported for variables such as weight. In contrast, 0 degrees Celsius does not actually imply "zero" temperature. It's meaningless to report the coefficient of variation for variables, such as degrees Celsius temperature.
When variables are expressed as logarithms, reporting the CV for these sets of variables becomes meaningless because a logarithm of 1 implies zero. This is because when the logarithmic scale is converted to another scale, the definition of zero would be redefined, and so would the value of CV. For example, it makes no sense to calculate the CV of a set of pH values, because pH is expressed in logarithmic scale, and pH does not actually mean "zero pH or no acidity."