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

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Basic Examples (6)

Log gives the natural logarithm (to base ):

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

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion shifted from the origin:

Asymptotic expansion at a singular point:

Scope (50)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Complex arguments:

Evaluate Log efficiently at high precision:

Log can deal with real‐valued intervals:

Log threads elementwise over lists and matrices:

It threads over lists in either argument:

Specific Values (5)

Simple exact values are generated automatically:

Values at infinity:

Zero argument gives a symbolic result:

Zero of Log:

Find a value of x for which the Log[x]=0.5:

Visualization (3)

Plot the Log function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties (12)

Log[z] gives the logarithm with base E:

Log is defined for all real positive values:

Complex domain:

Log achieves all real values:

The range for complex values:

Log is the inverse of Exp:

is not an analytic function:

Nor is it meromorphic:

The issue is a branch cut along the negative real axis:

The branch cut exists for any fixed value of :

is increasing on the positive reals for and decreasing for :

Log is injective:

Log is surjective:

Log is neither non-negative nor non-positive:

has both singularities and discontinuities for x≤0:

is concave on the positive reals for and convex for :

TraditionalForm formatting:

Differentiation (5)

The first derivative with respect to z:

The first derivative with respect to b:

Higher derivatives:

Formula for the derivative:

Derivative of a nested logarithmic function:

Integration (3)

Indefinite integrals of Log:

Definite integral of Log:

More integrals:

Series Expansions (5)

Taylor expansion for Log:

Plot the first three approximations for Log around :

General term in the series expansion of Log around :

Asymptotic expansions at the branch cut:

The first term in the Fourier series of Log:

Log can be applied to power series:

Function Identities and Simplifications (6)

Basic identity for Log:

Logarithm of a power function simplification:

Simplify logarithms with assumptions:

Logarithm of a product:

Change of base:

Expand assuming real variables x and y:

Function Representations (5)

Integral representation:

Series representation:

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)

Log can deal with real‐valued intervals from :

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:

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:

Neat Examples (6)

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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Log

Details

Examples

Basic Examples (6)