Built-in Mathematica Symbol

Log

Log[z]
gives the natural logarithm of z (logarithm to base e).

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

MORE INFORMATION

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 (3)

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

):

In[1]:=

Out[1]=

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

Scope (8)

Complex arguments:

Evaluate to high precision:

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

Log threads element-wise over lists:

Simple exact values are generated automatically:

Expand assuming real variables:

Find limits at branch cuts:

TraditionalForm formatting:

Generalizations & Extensions (5)

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:

Properties & Relations (15)

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:

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]]: