Math Is Fun Forum

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

log question

I visited the web site on numbers that ganesh mentions in This is Cool, I discovered...
There was a terrific technical writing on the movement of the moon.  But anyway, later
I found this and hoped someone could explain it.

Which of the following class-4 numbers is larger?

C = 2^2^2^83              (The ^ means to the power of.)
D = 3^3^3^52
as before we take the logarithm of both but this time we must do it twice, and we find

ln(ln(C)) = ln(ln(2)) + [ln(2) * 9671406556917033397649408]
= 6703708186976009930559261.24579...
ln(ln(D)) = ln(ln(3)) + [ln(3) * 6461081889226673298932241]
= 7098223961595389530659098.10481...
so D is larger.
(I see that 2^83 is the long 96714... number above, but I don't get how to get the log parts.)

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igloo myrtilles fourmis

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: log question

Hi John;

I am sure you know how now!

Tower powers are my favorite.
You are right, D is larger it is easier to see that by this:

while

Now you can see that D is the man by inspection.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: log question

I haven't solved a problem like this before so I was wondering how Bobbym got his answer.

Whew! we made it.

Taller towers would be even more complicated!

Last edited by Fruityloop (2013-08-10 18:29:26)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: log question

Hi Fruityloop;

That is fine work. But that is not how I do it. I am a very good guesser, watch this:

Cool hah!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

DaveRobinsonUK

Member

Registered: 2010-04-24

Posts: 123

Re: log question

Hi Guys

This is cool, always wondered how you would operate with Power Towers since Bobby first mentioned them to me. A while back now I know

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Can feel it coming together.. Slowly but Surely