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Log

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Log[z]
    gives the principal natural logarithm of z, with branch cut along
    the negative real axis.
Log[b, z]
    gives the logarithm to base b, i.e. Log[z] / Log[b].
Log is Listable. Log[1] = 0, Log[E] = 1, Log[E^n] = n for symbolic n.
Numeric inputs route to libm / MPFR; negative reals yield I Pi + Log[|z|].

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_log accepts 1 arg (natural log) or 2 (Log[b, z], base-b), emitting Log::argt otherwise. For Log[z]: exact special values Log[0] = -Infinity (exact), Log[0.] = Indeterminate, Log[1] = 0, Log[E] = 1, Log[Infinity] = Infinity; negative integers fold to I Pi + Log[-n] (the principal-branch shift). The key simplification is Log[E^k] -> k — but only when k is a real numeric (is_real_numeric_expr), because E > 0 keeps us on the principal branch and an unrestricted fold would cross the branch cut for complex k. Numeric inputs go through MPFR (mpfr_log for positive reals, a complex log|z| + I arg(z) helper otherwise) when USE_MPFR, else clog on a double complex, returning a real result only when the input is real-positive. For Log[b, z]: Log[b,b] = 1; Log[b, b^k] -> k under the same positive-base, real-k branch-cut guard (is_positive_known); exact integer powers (Log[2, 8] = 3) are found by repeated division; integer-zero argument resolves the directed ±Infinity from the sign of Log[b]. Everything else falls back to the rewrite Log[b, z] -> Log[z] / Log[b]. Unmatched inputs return NULL (left symbolic).

Data structures. Operates on Expr* trees; numeric paths use double complex (machine) or mpfr_t/the complex MPFR helpers (arbitrary precision). Negative-integer handling routes through GMP mpz_t to negate INT64_MIN/bigints safely.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Log[E]
Out[1]= 1

In[2]:= Log[E^2]
Out[2]= 2

In[3]:= Log[2, 8]
Out[3]= 3

In[4]:= Log[{1, E, E^2}]
Out[4]= {0, 1, 2}

In[1]:= Log[-1]
Out[1]= I Pi

In[1]:= D[Log[Sin[x]], x]
Out[1]= Cot[x]

In[1]:= Series[Log[1 + x], {x, 0, 6}]
Out[1]= x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + 1/5 x^5 - 1/6 x^6 + O[x]^7

In[1]:= N[Log[2], 40]
Out[1]= 0.69314718055994530941723212145817656807549

In[1]:= N[Log[1 + I], 40]
Out[1]= 0.34657359027997265470861606072908828403779 + 0.78539816339744830961566084581987572104928*I

Notes

Log[z] is the principal natural logarithm; Log[b, z] gives the base-b logarithm Log[z]/Log[b]. Log is Listable.