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Which is greater 4^300 or 3^400 ?

Guest May 10, 2014

Best Answer 

 #2
avatar+92673 
+13

Let's write

4300       ???      3400

(43)100  ???   (34)100

64100    ???       81100

And it's obvious that the thing on the right is greater than the thing on the left.

So.....

4300       <      3400

And there you go....

CPhill  May 10, 2014
 #1
avatar+3079 
+10

Definition:

$$x = log_b(a) \Leftrightarrow b^x = a \,\,\,\, (for \,\, a,b >0 \,\, and \,\, b<>1)$$

Formula to convert logarithm base g to base b:

$$log_b(x) = log_b(g) * log_g( x )$$

------------------------------------

4^300 = number "a"  is a really large number, but we don't care what the actual number is, instead we want to know how many digits the number has:

4^300 = a 

 300 = log4( a  )

and:

 log10( a) = log10( 4 ) * log4( a ) = log10( 4 ) * 300

 log10( a) = log10( 4 ) * 300

log10( a ) = 180.61799739838872 ( so about 180 digits)

------------------------------------

3^400 = b

 400 = log3( b  )

and:

 log10( b) = log10( 3 ) * log3( b ) = log10( 3 ) * 400

 log10( b) = log10( 3 ) * 400

log10( b ) = 190.848501887864976 ( so about 190 digits)

admin  May 10, 2014
 #2
avatar+92673 
+13
Best Answer

Let's write

4300       ???      3400

(43)100  ???   (34)100

64100    ???       81100

And it's obvious that the thing on the right is greater than the thing on the left.

So.....

4300       <      3400

And there you go....

CPhill  May 10, 2014
 #3
avatar+94105 
+5

Hi Chris,

That answer is really elegant.  I'm impressed!

 

Andre,

I have tried to work through yours - just in my head (which doesn't work very well)

But I don't get it.  Maybe I'll look again later.

Melody  May 10, 2014
 #4
avatar+92673 
0

Thanks....I actually didn't really know WHAT to do, at first. I was going to try something "complicated," but then I remembered a problem from a math "puzzles" book that I have that was similar to this. In fact, now that I think of it, I might post it to the forum just so some people can take a stab at it !!

CPhill  May 10, 2014
 #5
avatar+94105 
+5

That is an excellent idea.  You want to write or reference it somewhere in the puzzle threads.

 Reinout takes good care of them, and i really appreciate that,  but they could do with being organised a little better.  Maybe we should brain storm this one.  A lot of things need to be reorganised now that the forum has been revamped.

Melody  May 10, 2014
 #6
avatar+94105 
0

Hi Andre,

I have now looked at your answer properly.  It is really cool!

It is really great that one little  question got 2 totally different answers but both answers are really interesting and appropriate.

Melody  May 14, 2014

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