Exponential Equation

• December 30th 2012, 10:36 AM
Exponential Equation
Hi!
I got a pretty hard problem to solve:
${7^x} + {4^x} = {25^x}$

I simplified it with ${25^x}$ and I got:
${{{7^x}} \over {{{25}^x}}} + {{{4^x}} \over {{{25}^x}}} = 1$
from now on, in my textbook is written to use the monotony of the exponential function in order to solve this, the problem is that I don't really understand what they mean and I would really appreciate if you explain me!
• December 30th 2012, 10:48 AM
zachd77
Re: Exponential Equation
Monoticity of a function refers to whether the function is increasing, decreasing, or constant on an interval. It seems you are trying to solve this equation algebraically, that would be a pain. Is the textbook asking you to solve this algebraically or by means of graphing?
• December 30th 2012, 10:53 AM
Re: Exponential Equation
it is not specified, it just suggests to use the monoticity of the exponential function, and I know what monoticity of a function is, the promblem is that I don't really see how it reffers to my problem...
• December 30th 2012, 10:55 AM
zachd77
Re: Exponential Equation
Since the base of the exponent is greater than 1, that means the exponential function will always be increasing for all x. If you are trying to solve this equation, I'd suggest you graph the function asked and look for a point of intersection. Or, bring all terms onto one side of the equation, and find the root of that equation. Using algebra would be too messy and time consuming.
• December 30th 2012, 11:02 AM
HallsofIvy
Re: Exponential Equation
Is the problem to solve the equation or to determine whether or not there is a solution? With f(x)= (7/25)^x+ (4/25)^x, f(0)= 2> 1 and f()= 11/25< 1 so this equation clearly has a root, at approximately x= 1/2, but it is not a rational number and probably not any number that can be reasonably expressed exactly.
• December 30th 2012, 11:07 AM
Re: Exponential Equation
yes, the problem is to solve the equation, and there is a solution x=2, but I wanna know how to get to this solution mathematicaly
• December 30th 2012, 11:13 AM
Re: Exponential Equation
Attachment 26413
is this the way I should solve it?
• December 30th 2012, 11:20 AM
skeeter
Re: Exponential Equation
Quote:

yes, the problem is to solve the equation, and there is a solution x=2, but I wanna know how to get to this solution mathematicaly

then you must have copied the equation incorrectly ...

$7^2 + 4^2 \ne 25^2$
• December 30th 2012, 11:24 AM
${7^x} + {24^x} = {25^x}$