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# Log

 Log[z]gives the natural logarithm of z (logarithm to base e). Log[b, z]gives the logarithm to base b.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• Log gives exact rational number results when possible.
• For certain special arguments, Log automatically evaluates to exact values.
• Log can be evaluated to arbitrary numerical precision.
• Log automatically threads over lists.
• Log[z] has a branch cut discontinuity in the complex z plane running from to .
Log gives the natural logarithm (to base ):
Log[b, z] gives the logarithm to base b:
Series expansion:
Log gives the natural logarithm (to base ):
 Out[1]=

Log[b, z] gives the logarithm to base b:
 Out[1]=

Series expansion:
 Out[1]=
 Scope   (8)
Complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
Expand assuming real variables:
Find limits at branch cuts:
Log can deal with real-valued intervals from :
Zero and infinite arguments give symbolic results:
Log can be applied to power series:
Log threads over explicit lists as well as over sparse arrays:
Log is a numerical function:
 Applications   (8)
Plot Log for various bases:
Plot the real and imaginary parts of Log:
Plot the real and imaginary parts over the complex plane:
Plot data logarithmically and doubly logarithmically:
Benford's law predicts that the probability of the first digit is in many sequences:
Analyze the first digits of the following sequence:
Use Tally to count occurrences of each digit:
Shannon entropy for a set of probabilities:
Equi-entropy surfaces for four symbols:
Approximate the prime number:
Exponential divergence of two nearby trajectories for a quadratic map:
Compositions with the inverse function might need PowerExpand:
Get expansion that is correct for all complex arguments:
Simplify logarithms with assumptions:
Convert inverse trigonometric and hyperbolic functions into logarithms:
Log arises from the power function in a limit:
Solve a logarithmic equation:
Reduce a logarithmic equation:
Numerically find a root of a transcendental equation:
The natural logarithms of integers are transcendental:
Integrals:
Integral transforms:
Solve differential equations:
Limits:
Log is automatically returned as a special case for various special functions:
For a symbolic base, the base b log evaluates to a quotient of logarithms:
Generically :
Because intermediate results can be complex, approximate zeros can appear:
Machine-precision inputs can give numerically wrong answers on branch cuts:
Use arbitrary-precision arithmetic to obtain correct results:
Compositions of logarithms can give functions that are zero almost everywhere:
This function is a differential-algebraic constant:
Logarithmic branch cuts can occur without their corresponding branch point:
The argument of the logarithm never vanishes:
But it can take negative values, so the logarithm has a branch cut:
The kink at marks the appearance of the second sheet:
Logarithmic terms in Puiseux series are considered coefficients inside SeriesData:
In traditional form parentheses are needed around the argument:
Successive integrals of the log function:
Amoeba of a cubic:
Plot the Riemann surface of Log:
Plot Log at integer points:
Calculate Log through an analytically continued summed Taylor series:
Visualize how the value is approached as :
Plot the Riemann surface of Log[Log[z]]:
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