We start with:
#5^3-2x=5^(-x)#
and immediately run into a problem - in order to get the #x# out from the #5^(-x)# term, we need to use a log, but that will trap the #x# within the #2x# inside the log. And so the best way to solve this at this point is by graphing (the graphing utility is acting oddly when putting the two plots into the same grid, so I'll show them separately. This is the right side:
graph{(y-5^(-x))=0 [-20, 100, -30, 150]}
And this is the left:
graph{125-2x [-20, 100, -30, 150]}
And so by observation we can see two points of solution. One of them is roughly #x=-4# and the other is roughly #x=63#:
Testing the accuracy of #x=-4#
#5^3-2x=5^(-x)#
#5^3-2(-4)=5^(-(-4))#
#125+8=5^4#
#133=225# - meaning our answer is slightly less negative than #-4# (perhaps more like #-3.5#.) We could use a spread sheet to help get it closer.
Testing the accuracy of #x=63#
#5^3-2x=5^(-x)#
#5^3-2(63)=5^(-63)#
#125-126=1/(5^63)#
#-1="essentially 0"#
I think we can adjust the answer to roughly #x=62.5#
