This is my first time doing "log". So there is this question saying "evaluate. log5^5." What to do in here?? thanks!!

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This is my first time doing "log". So there is this question saying "evaluate. log5^5." What to do in here?? thanks!!

Mathematics
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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
So the answer is 5 log5? hm... my teacher wrote the answer on the whiteboard and she wrote it as 1. btw, 5 after log is down below. and then ^5.

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you mean like this? \[\log_{5}5 \]
yes. like that..
yes write, the question was log 5 (base 5) [cannot write it as the eqn editor is not working] so if you put it in the thing i wrote above it will come out to be 1
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.
@amogh oh.. ok.
@Keita: do you know what x^(1/2) means?
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?
okay good. does that make sense to you? you know that 4 squared is 16. you know that 4 cubed is 64 in other words, 4 squared is 4*4 4 cubed is 4*4*4 what is 4^2.5?
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
the lesson to be learned here is that any positive number can be expressed as a power of any other positive number.
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.
it so happens that any number can be expressed as a power of 3. go ahead and try 3^2.234234 and 3^ (any number you want)
logarithm is the inverse of this process. it means that log 32 = 2.5 in base 4. this means that 4^2.5 = 32 and log 15.5884573 = 2.5 in base 3. this means that 3^2.5 = 15.5884573
32 can also be expressed in base 10, or base 2. try lg 32 in google. that means log to the base 2 of 32 also try log 32. that means log to the base 10 of 32.
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
it's 32.0000001. yeah, I wrote that way cause I can't write the real one in my computer. so I put it as 5^5...
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?

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