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travis
 4 years ago
(logx)^(logx)=?
pfft didn't do log for quite a while :P
travis
 4 years ago
(logx)^(logx)=? pfft didn't do log for quite a while :P

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myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5\[\text{ Let } y=(\log(x))^{\log(x)} \] Now do log( ) of both sides. \[\log(y)=\log[(\log(x))^{\log(x)}]\] Using properties of log we can rewrite as \[\log(y)=\log(x)\cdot \log(\log(x))\] Again we can use another property to rewrite this as Now I will write this as an exponential equation where the base is 10 assuming the base is 10 here. \[10^{\log(y)}=10^{\log(x)\cdot \log(\log(x))}\] \[y=10^{\log(x) \cdot \log(\log(x))}\] => \[(\log(x))^{\log(x)}=10^{\log(x) \cdot \log(\log(x))}\] Both still pretty ugly, but I honestly don't know what you want to show here.

travis
 4 years ago
Best ResponseYou've already chosen the best response.0oh no, i have to proof its = x

travis
 4 years ago
Best ResponseYou've already chosen the best response.010 i think question didnt write

travis
 4 years ago
Best ResponseYou've already chosen the best response.0if it was another base im'm sure it wouldn't affect the answer

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5The base would just be some other number if not 10

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5\[(\log(x))^{\log(x)}=a^{\log(x) \cdot \log(\log(x))} \] if the base was a

travis
 4 years ago
Best ResponseYou've already chosen the best response.0Solve the equation (logx)^logx=x

travis
 4 years ago
Best ResponseYou've already chosen the best response.0thats how the question says

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5Oh we are solving an equation

travis
 4 years ago
Best ResponseYou've already chosen the best response.0log(logx)^logx=logx (logx)log(logx)=logx I cant cancel log on both sides right?

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5\[\log(x)(\log(\log(x))1)=0\]

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5subtract log(x) on both sides

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5both terms have log(x) in common

travis
 4 years ago
Best ResponseYou've already chosen the best response.0log(logx)=1 log(logx)=log10 logx=10 x=100 right?

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5what about x=10 for log(x)=0 since log(10)=1

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5you have two solutions

travis
 4 years ago
Best ResponseYou've already chosen the best response.0whats the other solution?

travis
 4 years ago
Best ResponseYou've already chosen the best response.0i took out logx so logx=0 right

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5\[(\log(x))^{\log(x)}=x\] D log( ) on both sides \[\log(x) \cdot \log(\log(x))=\log(x)\] Now subtract log(x) on both sides \[\log(x) \cdot \log(\log(x))\log(x)=0\] Now factor the expression on the left side \[\log(x)(\log(\log(x))1)=0\] Now set both factors =0 So we have \[\log(x)=0 \text{ or } \log(\log(x))1=0\] So we should solve both equations \[x=10 \text{ or } \log(\log(x))=1\] \[x=10 \text{ or } 10^{\log(\log(x))}=10^1\] \[x=10 \text{ or } \log(x)=10\] \[x=10 \text{ or } 10^{\log(x)}=10^{10}\] \[x=10 \text{ or } x=10^{10}\]

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5But if we check both solutions....

travis
 4 years ago
Best ResponseYou've already chosen the best response.0I divided the log away instead of subtracting it,now i get where i was wrong, thanks a lot that was very kind of you!

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.1I think you ment to say x=1 though after checking one sees that is not a solution

bahrom7893
 4 years ago
Best ResponseYou've already chosen the best response.0Travis you could have divided, you just made a mistake here: log(logx)=1 log(logx)=log10 logx=10 then x !=100

bahrom7893
 4 years ago
Best ResponseYou've already chosen the best response.010^log(x) = 10^10 x = 10^10

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5oh yeah zarkon for some rason i solve log(x)=1

travis
 4 years ago
Best ResponseYou've already chosen the best response.0Yea I was too busy manipulating the log I didn't convert the log properly

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5log(x)=0 x=1 for sure! :)

travis
 4 years ago
Best ResponseYou've already chosen the best response.0yea so x =1 and x= 10^10

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.1x=1 is only a solution if you define \[0^0=1\]

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5right which i don't define it that way

travis
 4 years ago
Best ResponseYou've already chosen the best response.0but if you sub x = 10^10 it works too

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5i would say that is the only solution

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.5because after checking x=1 we get 0^0 like zarkon said

travis
 4 years ago
Best ResponseYou've already chosen the best response.0So reject that solution right
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