If a transformation is applied on the data, there will be need to convert the results to the unit of the original data by applying the reverse of the transformation on the results. For instance if the logarithmic transformation is applied to make the data normally distributed, at the end of the day, the result will be converted using an exponential transformation back to the original unit. After differencing data to get it, for instance, stationary, you need to integrate the result back to the original data unit. It is therefore possible and allowed to use any transformation as long as you apply the reverse transformation on the result back to the original unit.

In my opinion, converting data, either linearly or non-linearly, changes the distribution of data and improve the accuracy of their measurement. But the interpretation of the converted data is different from the raw data, in which case their validity may be problematic.

"..., cautions should be taken when dealing with transformations that may affect the interpretation of the obtained model and subsequent statistical inference.(Huang 2018)." Harrell (2014) pointed out: “Playing with transformations distorts every part of statistical inference….”

Harrell F 2014 Comments on: ‘Pitfalls to avoid when transforming data?’ Cross Validated, http://stats.stackexchange.com/questions/90149/pitfalls-to-avoid-when-transforming-data

Huang H 2018 Uncertainty estimation with a small number of measurements, Part I: new insights on the t-interval method and its limitations Measurement Science and Technology 29 https://doi.org/10.1088/1361-6501/aa96c7

Usually in basic life sciences at least, transformations are used to linearize relationships between (transformed) variables and to get a residual distribution that is better fit by a normal distribution model, so that "standard" prodecures can be used to evaluate the information about the dependencies between the variables provided by the observed data.

The simplest proto-type case is the comparison of sample means in two groups (e.g. treated vs. control), where the t-test is applied. This test assumes that the residuals can be understood as a sample from a normal distribution*, and the hypothesis tested is that the expected difference of sample means is zero.

* the error term of the model can be formulated as a random variable with a normal distribution with mean 0 and a variance usually to be estimated from the observed data. Under this assumption, the response variable can be formulated as a random variable with a normal distribution with the mean depending on the predictros (the functional form of the statistical model) and constant variance (to be estimated).

Often in life sciences, the distribution of the response variable is approximated much better by a log-normal distribution than by a normal distribution. The distribution of the log response is then approximately normal. The t-test on the log response then test the hypothesis that the expected difference of mean log-values is zero, what is equivalent to test the hypothesis that the log ratio is zero or that the ratio is one. If the information from the observed data is considered sufficient to conclude non-zero log ratio, then it is equivalent to conclude a non-zero mean difference as well.

A real difference is that the model using the log-tranformed response estimates the mean log response, what is not the same as the log of the mean response. The model is actually working with the log of the geometric means thather than with arithmetic means. This should be considered when estimation is relevant.

Transformation of data is one thing that has become part of the entire process involved in scientific researches. We therefore get to understand that whenever data is transformed, it's validity (originality) or rather authenticity isn't altered. The big deal here is presenting the data itself using various techniques which also requires some transformation to be done on the same data. Therefore, transformed data in this case is still valid.