Çözüldü Exponential Equation

3^x + 4^x = 5^x
https://www.facebook.com/photo/?fbi...gm.525219756278520&idorvanity=158519972948502

Solution by 平凡人 :
First note that x = 2 is a solution by Pythagorean triple.
Then we prove that it is the only solution. Let f(x) = (3 / 5)^x + (4 / 5)^x - 1, then
f '(x) = [ (3 / 5)^x ]·ln(3 / 5) + [ (4 / 5)^x ]·ln(4 / 5)
Since ln(a) is negative when 0 < a < 1, thus f '(x) < 0 for all real x.
When x approaches infinity, f approaches -1.
When x approaches negative infinity, f approaches infinity.
Thus, f is strictly decreasing function and it follows that f has only one root. Therefore x = 2 is the only solution.

Another Solution Based On Intermediate Value Theorem (IVT)
f(x) = 3^x + 4^x - 5^x
x = 1 ⇒ f(1) = 2 > 0
x = 3 ⇒ f(3) = 27 + 64 - 125 = -34 < 0
Therefore, there's one root in (1, 3) which is x = 2.

Graph:

https://i72.servimg.com/u/f72/19/97/10/39/zstel15.png