Is the utility of money *actually* logarithmic? Apologies for asking what is probably a basic question from someone that is not in the economics field. 

But I was playing around with the idea of determining how a group of people could split a bill "fairly", meaning they each spend an equal amount of utility, calculated from their yearly income. And I did it using logarithms since that is the common recommendation.

I have seen many mentions that the utility of money is roughly logarithmic as it increases. And it is usually phrased as a rule of thumb, but I don't see actual justification. And the assumption seems embedded everywhere. For instance, a (real) flat income tax that taxes all income (not just consumption or earned income) is seen as neither regressive nor progressive - but the flat tax assumption (that everyone would pay the same percentage) again points to the idea of money having logarithmic utility.

My question is, has this actually been measured? I have seen indications that economists actually do studies where they analyze purchasing decisions across a wide range of income levels. It seems that with enough data points, a utility function could actually be calculated. 

Has this been done? And is it actually logarithmic? What is it? Is there an up to date utility function for the value of money?
"Essentially, all models are wrong, but some are useful"
George Box,  Empirical Model-Building and Response Surfaces

To what approximation would one like to know the utility of wealth? 

To be guaranteed that a utility function representation of preferences exists you need a number of assumptions about preferences to hold. Usually, these are Completeness and Transitivity with Continuity, Monotonicity, and Convexity common additional assumptions to give the nicely behaved preferences we see in most economics settings. Levin  and Milgrom (2004) provides a technical, but short overview of the definitions and derivations in this branch of economics called "Choice Theory".

But that's just for ordinal utility functions (functions which rank bundles of goods but resulting the utils have no economic meaning beyond that a bundle with higher utils is preferred to one with lower utils) to exist. It turns out that the preferences of ordinal utility functions are preserved by any positive affine transformation. To get a particular functional form like $log(x)$ to matter you need cardinal utility where the values of the preferences matter (although even here they can be re-scaled by a positive constant and a constant added). Cardinal utility functions are more common in finance and macroeconomic settings even though more is required for such a utility function to exist. As Wikipedia says:


  The idea of cardinal utility is considered outdated except for
  specific contexts such as decision making under risk, utilitarian
  welfare evaluations, and discounted utilities for intertemporal
  evaluations where it is still applied. Elsewhere, such as in
  general consumer theory, ordinal utility with its weaker assumptions
  is preferred because results that are just as strong can be derived.


Cardinal utility 

Given the findings of behavioral economics experiments, which provide evidence that preferences may violate completeness and transitive axioms, it is likely that no such "true" function exists. However, as Box's quote indicates, they may still be useful. 

The log utility function has nice properties. The logarithmic utility function is a special case of constant relative risk aversion (CRRA) utility function. Roughly speaking, this family of utility functions views risks in percents of wealth as constant for all levels of wealth. That is, rich and poor alike worry the same about a 10% shock to wealth. Equivalently, the utility "pain" of spending 10% of wealth on something are the same at all wealth levels. This is an example of a function with a declining marginal utility of consumption.  

Log utility is particularly easy to work with because it has simple derivatives and reflects a "low" but non-zero level of preferences over risk. Also, under log utility the wealth and substitution effects of interest rates cancel out in intertemporal choice problems, which further simplifies some models. In portfolio allocation, log utility maximization is effectively maximizing the geometric mean return the return in the long run, which feels intuitively sensible as an investment goal for the long run investor. Also, when working with shocks to wealth that are log-normal and a log utility function, there are nice closed form solutions to the values of expected  utility (they are slightly more complex in the CRRA case). Here is an example of trying to calibrate CRRA utility using experimental data:


  For this specification of CRRA, a value of 0 denotes risk neutrality,
  negative values indicate risk-loving, and positive values indicate
  risk aversion.  Thus we see clear evidence of risk aversion: the mean
  CRRA coefficient is 0.64. This distribution is consistent with
  comparable estimates obtained in the United States, using college
  students and an MPL design, by Holt and Laury [2002] and Harrison,
  Johnson, McInnes and Rutström [2003a][2003b].


Harrison, Lau, Rutstrom (2007)

If CRRA were the proper specification and the experimental point estimates recovered the true CRRA parameter, then we would be close to log utility which occurs when you take the limit of a CRRA utility function as the coefficient approaches $1$. 

In conclusion, there is some empirical evidence that preferences can be approximated with a log utility function and it is very easy to work with.