Solve for t over the real numbers:
5000 = 2^(5 t) 5^(5 t+2)
5000 = 2^(5 t) 5^(5 t+2) is equivalent to 2^(5 t) 5^(5 t+2) = 5000:
2^(5 t) 5^(5 t+2) = 5000
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):
5 log(2) t+log(5) (5 t+2) = log(5000)
Expand and collect in terms of t:
(5 log(2)+5 log(5)) t+2 log(5) = log(5000)
Subtract 2 log(5) from both sides:
(5 log(2)+5 log(5)) t = log(5000)-2 log(5)
Divide both sides by 5 log(2)+5 log(5):
Answer: | t = (log(5000)-2 log(5))/(5 log(2)+5 log(5)) =Log(200)/5
