#1**+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:

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

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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)

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**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**+13 **

Best Answer

Let's write

4^{300} ??? 3^{400}

(4^{3})^{100} ??? (3^{4})^{100}

64^{100} ??? 81^{100}

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

So.....

4^{300} < 3^{400 }

And there you go....

CPhill
May 10, 2014

#3**+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**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**+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