Linear models on a log transformed response model relative (rather than absoluet) changes on the original response. The estimates, however, refer to the expectation of logarithms. Since this is a monotone transformation, the direction of changes is the same on the scale of the original response, but getting estimates for the magnitude of the effets on the original scale is a bit complicated (see https://www.statlect.com/probability-distributions/log-normal-distribution)

This note on UCLA's IDRE site may be helpful.

• https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faqhow-do-i-interpret-a-regression-model-when-some-variables-are-log-transformed/

Mulatu - 

If you are looking at linear regression with skewed response and predictor data, then weighted least squares (WLS) regression works very well.  If you are using logs to reduce heteroscedasticity in such a case, then that just artificially addresses a natural phenomenon, more or less, which should be addressed in the error structure.  See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as bounded by Ken Brewer.  

The following presentation contains references to such cases where WLS regression worked very well for ratio estimators:

Invited presentation for mathematical statisticians at the US Energy Information Administration:   

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development, using prediction. 

This also works and can be helpful in more complex cases:

https://www.researchgate.net/project/Cutoff-and-quasi-cutoff-sampling-with-prediction-for-Official-Statistics.  Also please see the updates which appear in reverse chronological order. 

Cheers - Jim

I. Power laws possess the attribute of scale invariance. From that follows a straight-line log–log plot, what often helps to identify a power law trend. Power laws often provide fairly good correlations with a minimal number of parameters. Dimensionless quantities are often correlated by means of empirical correlations that often follow (or closely follow) power laws.

II. Example in the application of dimensional analysis to transport phenomena (engineering) ― For forced convection, we may quite generally express the Nusselt number as a function of both the Reynolds and the Prandtl numbers for great diversity of conditions, namely geometric, with simply 3 parameters: Nu = k·Rep·Prq. It may be even possible to further reduce the number of degrees of freedom by means of a predetermined constant (possibly the unit: p + q = cte. = 1), to impart a compensated 'weight' to each contribution. It may be convenient to take the parameters as time varied.

III. This very limited number of degrees of freedom may contribute to stabilize numerical procedures, namely those related with real-time parameter estimation, with real-time optimization, and with adaptive control; particularly if processes occurs concurrently, and if no prior 'learning phase' is reserved for just parameter estimation. Imposed (limited) disturbances may also play a stabilizing role. Both real-time optimization and with adaptive control may be suited to benefit from concurrent parameter estimation.

IV. The least-squares fitted parameters obtained after the logarithmic-linearized correlation are biased for the exponential correlation, but typically close to the unbiased estimates if obtained directly from the original power-law correlation.

V. The recursive least squares algorithm (RLS) allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. RLS is the recursive application of the well-known LS regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. For RLS with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data. No prior 'learning phase' is required.

VI. Application example ― While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. The power dissipated by agitation was accessed by a torque meter (pilot plant). Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...

Example that may be of interest:

One dataset I think I considered for   https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity was the home sales price data shown in Example 13-3 in https://online.stat.psu.edu/stat501/lesson/13/13.2.  Penn State provides a lot of good information, and interesting examples. The residual versus fitted graph there shows obvious heteroscedasticity.  However, they state the following: "Since all the variables are highly skewed we first transform each variable to its natural logarithm."  Then "response is logY."  So this log transformation likely substantially lowered the coefficient of heteroscedasticity for the case studied, but it was not enough to account for the heteroscedasticity actually present.  ( For more on estimating the coefficient of heteroscedasticity see https://www.researchgate.net/publication/354854317_WHEN_WOULD_HETEROSCEDASTICITY_IN_REGRESSION_OCCUR.)

That example considered on that Penn State site is quite interesting. 

There is no merit. See why: www.lerenisplezant.be/fitting.htm