+3144 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)
+128406 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....
+118608 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.
+128406 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 !!
+118608 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.
