Logarithm

Author

Eric W. Weisstein
September 8, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Logarithm.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Logarithm.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Graphics`Colors`

Plot[{Log[x],Log[2,x],Log[10,x]},{x,.001,10},
AxesLabel->TraditionalForm/@{x,log[b,x]},
PlotRange->{Automatic,{-2.4,2.4}},PlotStyle->List/@{Black,Blue,Red}]

<<Utilities`Plot`

ComplexPlot3D[Log[z],{z,-5,5},ComplexPlotLabel->log[E,z]]

ComplexContourPlot[Log[z],{z,-5,5},ComplexPlotLabel->log[E,z]]

Derivative

D[Log[b,z],z]

Integral

Simplify[Integrate[Log[b,z],z]]

Identities

{a^(Log[a,b]/Log[a,b]),(a^Log[a,b])^(1/Log[a,b]),b^(1/Log[a,b])}

{Log[b,a],1/Log[a,b]}

{Log[b,x],Log[b,y^Log[y,x]],Log[y,x]Log[b,y]}

{a^x,b^(x/Log[a,b]),b^(x Log[b,a])}

Numerator/Denominator Identity

id=Log[((a+Sqrt[a^2-c^2])/c)]-1/2 Log[((a+Sqrt[a^2-c^2])/(a-Sqrt[a^2-c^2]))]

FullSimplify[id,0<c<a]

Table[id/.Thread[{c,a}->Sort[Table[Random[Real,10],{2}]]],{100}]//Chop//Union

FullSimplify[1/2 Log[((a+Sqrt[a^2-c^2])/(a-Sqrt[a^2-c^2]))],0<c<a]

FullSimplify[Expand[(((((a+Sqrt[a^2-c^2])/c)))^2)-((a+Sqrt[a^2-c^2])/(a-Sqrt[a^2-c^2]))]]

Simplification of Logs

d[a_]:=Times@@Select[a^(1/Range[Ceiling[Log[2,a]]]),IntegerQ]
U[1]:=1
U[n_]:=U[n]=Plus@@(Log[d[#]]U[n/#]&/@Rest[Divisors[n]])/Log[n]

U/@Range[10]

LogSimplify/@%