Solve for x :
2 5^(x + 2) = 5 2^(x + 5)
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(5) (x + 2) + log(2) = log(2) (x + 5) + log(5)
Expand out terms of the left hand side:
log(5) x + log(2) + 2 log(5) = log(2) (x + 5) + log(5)
Expand out terms of the right hand side:
log(5) x + log(2) + 2 log(5) = log(2) x + 5 log(2) + log(5)
Subtract x log(2) + log(2) + 2 log(5) from both sides:
(log(5) - log(2)) x = 4 log(2) - log(5)
Divide both sides by log(5) - log(2):
x = (4 log(2) - log(5))/(log(5) - log(2))=1.2694123......
