Science Forums 2

[CraigD]

Originally Posted by Don Blazys

To: CraigD. P.S. In LaTex, the logarithmic exponent should assume the form…

Mea maxima culpa! My apologies for taking so long to get it right. All those parentheses got me mixed up – embarrassing for a guy who writes parsers and other parentheses-y things for a living. Hopefully, we’ve now gotten past the electronic barrier of writing the equation (which would have taken all of 30 seconds using an old-fashioned chalkboard). Though, to spread the culpa around a bit, Don could have saved me from my folly by learning a bit of LaTeX. It’s not difficult, and offers the pleasure at seeing ugly keystrokes rendered as pretty pictures.

Here’s the equation correctly rendered:


A bit more compactly:


Dividing both sides by , this can be written

I’m not sure if this is what the Don meant in the original post by “cross out the canceled Ts”. I don’t see anything algebraically abnormal about the equation

It is a remarkable identity, and looks like fun to prove. It’s pretty easy to demonstrate, running a little MUMPS program (with a standard, 18-digit precision calculator) on my cheap 4-year-old laptop (no supercomputer required ):

f  r "a=",a," T=",T," x=",x," " s E=x*$zln(a)/$zln(T)-1/($zln(a)/$zln(T)-1) w a,"^",x,"=",a**x," ",T,"(",a,"/",T,")^(",E,")=",T*(a/T**E),!
a=1 T=2 x=3 1^3=1 2(1/2)^(1)=1
a=2 T=3 x=4 2^4=16 3(2/3)^(-4.128533874054364329)=15.99999999999999945
a=2 T=4 x=3 2^3=8 4(2/4)^(-1)=8
a=4 T=2 x=3 4^3=64 2(4/2)^(5.000000000000000001)=63.9999999999938609
a=9 T=3 x=2 9^2=81 3(9/3)^(2.999999999999999998)=80.99999999999948382
a=16 T=4 x=2 16^2=256 4(16/4)^(3)=256
a=25 T=5 x=2 25^2=625 5(25/5)^(3)=625
a=8 T=2 x=2 8^2=64 2(8/2)^(2.5)=63.99999999999999652
a=.25 T=.5 x=2 .25^2=.0625 .5(.25/.5)^(3.000000000000000001)=.0624999999999969838
a=.5 T=.6 x=.7 .5^.7=.6155722066724512082 .6(.5/.6)^(-.1405352050771790835)=.6155722066724661252
a=.5 T=3 x=2 .5^2=.25 3(.5/3)^(1.386852807234541587)=.2499999999999999888
a=2.718281828459045236 T=7.389056098930650228 x=2 2.718281828459045236^2=7.389056098930650231 7.389056098930650228(2.718281828459045236/7.389056098930650228)^(-.000000000000000002)=7.389056098930649799
a=7.389056098930650228 T=2.718281828459045236 x=3 7.389056098930650228^3=403.4287934927351227 2.718281828459045236(7.389056098930650228/2.718281828459045236)^(5.000000000000000002)=403.4287934927350995

The difference between the two sides of the equation at more than 18 significant digits are typical of rounding error when approximating transcendental functions such as logarithms, which are used not only in the natural log focution (, $zln), but also for non-integer exponentiation.