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:

Log can deal with real‐valued intervals from :

Log is a numerical function:

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:

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]]: