"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The idea of cardinal utility is considered outdated except for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.
