Log
Log[z]
gives the natural logarithm of z (logarithm to base ).
Log[b,z]
gives the logarithm to base b.
Details

- Log is a 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
.
Examples
open allclose allBasic Examples (6)
Scope (44)
Numerical Evaluation (6)
Specific Values (5)
Visualization (3)
Function Properties (6)
Differentiation (5)
Series Expansions (5)
Function Identities and Simplifications (6)
Basic identity for Log:
Logarithm of a power function simplification:
Function Representations (5)
Log arises from the power function in a limit:
Log can be represented in terms of MeijerG:
Log can be represented as a DifferentialRoot:
Generalizations & Extensions (2)
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:
Properties & Relations (13)
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:
Integral transforms:
Solve differential equations:
Limits:
Log is automatically returned as a special case for various special functions:
Possible Issues (7)
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:
Text
Wolfram Research (1988), Log, Wolfram Language function, https://reference.wolfram.com/language/ref/Log.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "Log." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Log.html.
APA
Wolfram Language. (1988). Log. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Log.html