Suppose two independent variables in the linear regression initially have very high correlation of 0.95. This introduces severe multicollinearity into the model (as indicated by very high variance inflation factors). Can one take natural logarithm of each of them (this decreases correlation between them to 0.75), and use them in the same regression? VIFs do not indicate multicollinearity issues then. Is it a reasonable approach?
Sometimes variables are just correlated. It's not necessarily bad, it's just the way that it is.
If you're doing regression, controlling for one variable, that's because you want to control for it. If you do a log transformation, you'll alter the meaning of the variables, you'll alter the distributions of the residuals, and you might introduce non-linear effects. If that all makes sense, go ahead and log them.
In addition, if log transformation reduces the correlation that much, they must have slightly strange distributions. Again, that might be OK, but it might also be something you'll want to look into.
Your results are not entirely surprising. Many variables that represent time series are highly correlated by nature (everything grows over time). Such variables are not only multicollinear to each other. But, they also render regression models heteroskedastic and resulting residuals are autocorrelated (not random). All those issues result into really flawed regression models that can't properly determine the statistical significance of any independent variables used in such a model.
When you transformed the variables to logs, the correlation decreased quite a bit. That's a good thing in most instances. Although taking the log may not entirely rid your regression models of the other potential problems I have mentioned.
You may also want to try converting your independent variables to a % change from one period to another instead of taking the log or just keeping the nominal value as in your original data. When doing so, you should typically see the correlation between the two independent variables drop even a lot more than what you observed when taking logs. Lower correlation between independent variables is good.