Step-by-step explanation:
I think you made some typos in the problem definition.
particularly the function definition is wrong.
the correct function to calculate the frequency for every key of a piano keyboard is
f(n) = 440 Ă— 2^((n-49)/12)
now we have a given frequency and do the reverse function on it to find the key number with that frequency.
55 = 440 Ă— 2^((n-49)/12)
55/440 = 2^((n-49)/12)
log2(55/440) = (n-49)/12
12Ă—log2(55/440) = n - 49
n = 12Ă—log2(55/440) + 49 = 12Ă—log2(1/8) + 49 =
= 12Ă—-3 + 49 = -36 + 49 = 13
the frequency of the 13th key is 55 Hz.
f(13) = 55 Hz.
about log2(1/8) :
remember that 1/(x^n) can also be written as x^(-n).
the 1/... operation creates a negative exponent.
8 = 2Âł
so, 1/8 = 2^(-3)
and therefore log2(2^(-3)) = -3