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log a^x = x times log a
so log 5^5 = 5 x log 5 = 5 log5

Greetings Keita,
If y=a^x
then log y (base a) = x

you mean like this?
\[\log_{5}5 \]

yes. like that..

oh. then the answer is 1.

Log of anything to to the base of itself it 1

do one thing. copy and paste
4^2.5 in google and tell me what the answer is.

ok I'll google it. @ amogh, so how to solve it? do you know?

I guess dhatraditya is proceeding correctly.

@ dhatraditya it's equal to 32.

okay good. the answer is 32. but what does 4^2.5 mean? can you say it out in english?

4 to the power of 2.5? x to the power of 1/2?

hm... no idea ...

4^2 = 4^1 * 4^1 correct?

yup..

so 4^2 = 4^1*4^1 = 4^(1+1)
similarly 4^2.5 = 4^1 * 4^1 * 4^0.5

but what do we mean by 4^0.5? we mean 4^(1/2) = square root of 4 = 2
so 4^2.5 = 4*4*2 = 32

go ahead and try 3^2.5 in google

it's 15.5884573.

ah, now try 3*3*sqrt 3

um.. it's same answer. 15.5884573.

great! so do you see a pattern here? we were able to express the number 15.5884573 as a power of 3.

log 32=1.50514998 in google. which is near to 2. so, log 5^5? how to do this?

so in your problem
\[\log_{5}5 \]
it means log 5 to the base 5.

log5^5 has a different meaning. it means log 5 raised to the power 5.

anyway, try 10^1.50514998 in google

okay. do you see a relationship here?
log 32(to the base 10) = 1.505
and 10^1.505 = 32.

so lets say log 5 to the base 5 = x.
what must this mean?

yup, I understand that now..

hm... it means 5 is expressed in base 5. So that means log to the base 5 of 5??

log5(5) = y :means: 5^y = 5
What does 'y' have to equal for this to be true?

hmm. 1. bcoz 5^1 is 5, isn't it?