Question

What is the inverse of `log (x/2)` ?

Answer 1

Assuming this is base-10 logarithm, the inverse function is
`y=2*10^x`

Explanation:

Function `y=g(x)` is called inversed to function `y=f(x)` if and only if
`g(f(x))=x` and `f(g(x))=x`

Just as a refreshment on logarithms, the definition is:
`log_b(a)=c` (for `a>0` and `b>0`)
if and only if `a=b^c`.
Here `b` is called a base of a logarithm, `a` - its argument and `c` - its balue.

This particular problem uses `log()` without explicit specification of the base, in which case, traditionally, base-10 is implied. Otherwise the notation `log_2()` would be used for base-2 logarithms and `ln()` would be used for base-`e` (natural) logarithms.

When `f(x)=log(x/2)` and `g(x)=2*10^x` we have:
`g(f(x))=2*10^(log(x/2))=2*x/2=x`
`f(g(x))=log((2*10^x)/2)=log(10^x)=x`