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

- anonymous

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- anonymous

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

- anonymous

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

- anonymous

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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- anonymous

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

- anonymous

yes. like that..

- anonymous

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

- anonymous

oh. then the answer is 1.

- anonymous

Log of anything to to the base of itself it 1

- anonymous

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

- anonymous

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

- anonymous

I guess dhatraditya is proceeding correctly.

- anonymous

@ dhatraditya it's equal to 32.

- anonymous

@amogh oh.. ok.

- anonymous

@Keita: do you know what x^(1/2) means?

- anonymous

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

- anonymous

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

- anonymous

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?

- anonymous

hm... no idea ...

- anonymous

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

- anonymous

yup..

- anonymous

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

- anonymous

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

- anonymous

the lesson to be learned here is that any positive number can be expressed as a power of any other positive number.

- anonymous

go ahead and try 3^2.5 in google

- anonymous

it's 15.5884573.

- anonymous

ah, now try 3*3*sqrt 3

- anonymous

um.. it's same answer. 15.5884573.

- anonymous

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

- anonymous

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)

- anonymous

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

- anonymous

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.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

anyway, try 10^1.50514998 in google

- anonymous

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...

- anonymous

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

- anonymous

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

- anonymous

yup, I understand that now..

- anonymous

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

- amistre64

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

- anonymous

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

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