10.5 = 3.5 1.07^n
10.5 = 21/2 and 3.5 1.07^n = 7 2^(-1-2 n) (107/25)^n:
21/2 = 7 2^(-1-2 n) (107/25)^n
21/2 = 7 2^(-1-2 n) (107/25)^n is equivalent to 7 2^(-1-2 n) (107/25)^n = 21/2:
7 2^(-1-2 n) (107/25)^n = 21/2
Divide both sides by 7:
2^(-1-2 n) (107/25)^n = 3/2
Take the natural logarithm of both sides and use the identities log(a b) = log(a)+log(b) and log(a^b) = b log(a):
log(2) (-1-2 n)+log(107/25) n = log(3/2)
Expand and collect in terms of n:
(log(107/25)-2 log(2)) n-log(2) = log(3/2)
Add log(2) to both sides:
(log(107/25)-2 log(2)) n = log(3/2)+log(2)
Divide both sides by log(107/25)-2 log(2):
Answer: | n = (log(3/2)+log(2))/(log(107/25)-2 log(2)=16.238
