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# Which is greater 4^300 or 3^400 ?

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Which is greater 4^300 or 3^400 ?

Guest May 10, 2014

#2
+80823
+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
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#1
+3066
+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)

#2
+80823
+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
#3
+91412
+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
+80823
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
+91412
+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
+91412
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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