The roots of equation 3^x - 4^x = 0 is
a) -1
b) 0
c) 1
d) 2
Answers
One direct method is to divide directly by 5x and get 1=(3/5)x+(4/5)x. From here it is clear that the RHS is strictly decreasing, and there is a unique solution. Almost all exponential equations can be treated this way, by transforming them to
one increasing function equal to one decreasing function
one increasing/decreasing function equal to a constant.
If we insert the known solution we can write
52+x=42+x+32+x
asking, whether there might another solution exist besides x=0 . Then we can rewrite, putting the 5x to the rhs:
52=42⋅0.8x+32⋅0.6x
Then if the exponents x on the rhs are zero, we have the known solution. But if x increases over zero, then the values of both summands decrease simultaneously, thus the equality can no more hold.
The analogue occurs for decreasing x: both summands increase over their squares simultaneously, so there is no other solution possible. QED.