exp equation

\hspace{-16}$Find real values of $\bf{x}$ in $\bf{ {5}^{1+x}+{5}^{1-x}={2}^{1+x}+{2}^{1-x}+{3}^{1+x}+{3}^{1-x}}$

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36
rahul ·

i am just writing a soln.. but don't know weather its right or wrong..

clearly, 21+x + 21-x + 31+x + 31-x are all positive and real.. for real x .. so why not A.M-G.M

so, we get, 51+x + 51-x > (36)1/4

or, 5x + 5-x > 61/2/5

or, 5x > 61/2/5 - 5-x

=> x > log5(61/2/5 - 5-x)

comparison can be made only if the terms are real...

and for R.H.S to be real...
the definiton of log should be used...

i.e, 5-x < 61/2/5

again, taking logs...
i know its too much but there's no harm if are confirmed that the terms are positive and also the bases i.e. greater than 0 less than 1 or greater than 1...

=> -x < log561/2 - 1

=> x > 1 - log561/2

this is the x for which we got x to be defined for...

but i don't think that this the answer as well...

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