Solve for m over the integers:
3^(5 m+14) (2^m)^m = 2^(8 m-7) (3^m)^m
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) m^2+log(3) (5 m+14) = log(2) (8 m-7)+log(3) m^2
Expand out terms of the left hand side:
5 log(3) m+log(2) m^2+14 log(3) = log(2) (8 m-7)+log(3) m^2
Expand out terms of the right hand side:
5 log(3) m+log(2) m^2+14 log(3) = log(3) m^2+8 log(2) m-7 log(2)
Subtract m^2 log(3)+8 m log(2)-7 log(2) from both sides:
-log(3) m^2+5 log(3) m+log(2) m^2-8 log(2) m+7 log(2)+14 log(3) = 0
Collect in terms of m:
(5 log(3)-8 log(2)) m+(log(2)-log(3)) m^2+7 log(2)+14 log(3) = 0
The left hand side factors into a product with two terms:
(m-7) (-log(3) m+log(2) m-log(2)-2 log(3)) = 0
Split into two equations:
m-7 = 0 or -log(3) m+log(2) m-log(2)-2 log(3) = 0
Add 7 to both sides:
m = 7 or -log(3) m+log(2) m-log(2)-2 log(3) = 0
Collect in terms of m:
m = 7 or (log(2)-log(3)) m-log(2)-2 log(3) = 0
Add log(2)+2 log(3) to both sides:
m = 7 or (log(2)-log(3)) m = log(2)+2 log(3)
Divide both sides by log(2)-log(3):
m = 7 or m = (log(2)+2 log(3))/(log(2)-log(3))
The root m = (log(2)+2 log(3))/(log(2)-log(3)) is not an integer:
Answer: |m = 7
