Hello,
I'm trying to regress a dependent variable against two independent variables using a linear model. However, since the estimated coefficients are very small, I thought about log-transforming the variables in order to get a better result.
My model: lm <- lm(x ~ a+b)
Now my question is, if I have to perform the transformation on both the dependent variable and the independent variables ( lm(log(x) ~ log(a)+log(b)) ) or only on the independent variables ( lm(x ~ log(a)+log(b)) ).

I'm sorry, but your response doesn't seem to me to argue a position. Being close to zero is saying that its 'small' with different words, and as I don't understand what the problem is with having small numbers, neither do I understand the problem with having numbers that are close to zero...

I want to convince you that small numbers, close to zero, on some scale, are not inherently problematic. I would like you to try to convince me that they are, or that at least you have a special case in which they are are a problem given your context.

5e-06 meters is 5000 nano-meters.
is 5000 close to zero ?

In case you want to interpret the results (e.g.: By stating that an increase of one unit in the independent variable causes y to increase by x units) larger numbers are just more pleasant to use in my opinion.
You are right that the small numbers would work too and it wouldn't make much of a difference.

However, my main question is, how the log-transformation would work in this case. Is it necessary to apply the log() to both the dependent and independent variables or is it also correct to apply to log() only to the independent variables?