which log operations are involved in this formula?

Question

I've a function f1(x) that "scale" (not sure how to call it) a power of 2 function, such as:

Now, from this value, I display the dB measure (in the form log10(x) * 20; usually used for audio and stuff):

The fact is: it works if I also multiply it for a (i.e. log10(x) * 20 * a). If I remove it, the output is wrong (i.e. 6db for each twice).

Not sure the operations that simply permit to return correctly the result multipling by a.

Any smart log's operations involved which I don't see?

Answer 1

Basically the "trick" here is that $\log(x^{a}) = a\log(x)$.