When can taking the following transformation of data $log_{10}(e^{mean(log(x))})$ be useful? I wished to transform some data and have noticed that the geometric mean is a known transformation to normalise data with high differences in comparably large values relative to lower values.
However, has such a transformation been used before:
$log_{10}(e^{mean(log(x))})$
When does using the $log_{10}$ of the geometric mean become meaningful? for example, if I have a set of values and I want to take the differences likes:
x <- data.frame(m = rep(1:2, 2),x=c(46, 7, 888, 9), y=c(888, 6, 77, 5), d=c(999, 66, 5, 4))

#geometric mean
ge <- function(x){
+     exp(mean(log(x)))
+ }

#log10 mean
ge <- function(x){
+     log10(exp(mean(log(x))))
+ }

#output
> aggregate(. ~ m, x, ge)
  m          x          y        d
1 1 202.108882 261.488049 70.67531
2 2   7.937254   5.477226 16.24808
> aggregate(. ~ m, x, log.ge)
  m         x         y        d
1 1 2.3055854 2.4174518 1.849268
2 2 0.8996703 0.7385606 1.210802

If I were to take the differences from the rows of 2 from 1, by ratio it is far larger in the geometric mean in comparison to taking the $log_{10}$ how much does this impact the data?

Introducing the log is equivalent to scaling the (non-exponentiated) mean by $\frac{1}{\log_e(10)}$. Notice that you can change the base of a logarithm from $a$ to $b$ by scaling:
\begin{align}
\log_a{x} = \frac{\log_b(x)}{\log_b(a)}
\end{align}
We therefore have:
\begin{align}
\log_{10}\left(e^{\text{mean}(\log(x))}\right)
  & = \frac{1}{\log_{e}(10)}\log_e\left(e^{\text{mean}(\log(x))}\right) \\
  & = \frac{1}{\log_{e}(10)}\text{mean}(\log(x))
\end{align}
