Solve for n over the real numbers:
3000 = 6000 0.85^n
6000 0.85^n = 3 4^(2-n) 5^(3-n) 17^n:
3000 = 3 4^(2-n) 5^(3-n) 17^n
3000 = 3 4^(2-n) 5^(3-n) 17^n is equivalent to 3 4^(2-n) 5^(3-n) 17^n = 3000:
3 4^(2-n) 5^(3-n) 17^n = 3000
Divide both sides by 3:
4^(2-n) 5^(3-n) 17^n = 1000
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):
2 log(2) (2-n)+log(5) (3-n)+log(17) n = log(1000)
Expand and collect in terms of n:
(-2 log(2)-log(5)+log(17)) n+4 log(2)+3 log(5) = log(1000)
Subtract 4 log(2)+3 log(5) from both sides:
(log(17)+(-2 log(2)-log(5))) n = log(1000)+(-4 log(2)-3 log(5))
Divide both sides by -2 log(2)-log(5)+log(17):
Answer: | n = (-4 log(2)-3 log(5)+log(1000))/(-2 log(2)-log(5)+log(17))=4.2650242818.......etc.
