A Very Interesting Exponential Math Olympiad Question | Find x

[@RealQinnMalloryu4]

{10^3+10^5}=20^5 10^10^5 2^5^2^5^5 1^1^2^1^1 21 (x ➖ 2x+1).

[@ManojkantSamal]

^=read as to the power
*=read as square root
10^x+100^x=1000^x
(10)^x+{10^x}^2={10^x}^3
Let 10^x=t
So,
t+t^2=t^3
t=t^3-t^2
t=t(t^2-t)
t^2-t=t/t=1
t^2-t-1=0
t=-(-1)±*{(-1)^2-4.1(-1)}/2.1
={(1)±*(1+4)}/2
=(1±*5)/2
-ve is not accepted
So,
t=(*5+1)/2
10^x=(*5+1)/2
Take the log
log 10^x= log {(*5+1)/2}
X. log 10 ={log (*5+1) - log 2}/log 10