# Function LOG

### Syntax

• log number `&optional` baselogarithm

### Description

log returns the logarithm of number in base base. If base is not supplied its value is `e`, the base of the natural logarithms.

log may return a complex when given a real negative number.

```
(log -1.0) ≡ (complex 0.0 (float pi 0.0))

```

If base is zero, log returns zero.

The result of `(log 8 2)` may be either `3` or `3.0`, depending on the implementation. An implementation can use floating-point calculations even if an exact integer result is possible.

The branch cut for the logarithm function of one argument (natural logarithm) lies along the negative real axis, continuous with quadrant II. The domain excludes the origin.

The mathematical definition of a complex logarithm is as follows, whether or not minus zero is supported by the implementation:

```
(log `x`) ≡ (complex (log (abs `x`)) (phase `x`))

```

Therefore the range of the one-argument logarithm function is that strip of the complex plane containing numbers with imaginary parts between `-π` (exclusive) and `π` (inclusive) if minus zero is not supported, or `-π` (inclusive) and~`π` (inclusive) if minus zero is supported.

The two-argument logarithm function is defined as:

```
(log base number) ≡ (/ (log number) (log base))

```

This defines the principal values precisely. The range of the two-argument logarithm function is the entire complex plane.

### Examples

```
(log 100 10)

→
2.0
or 2

(log 100.0 10)

→
2.0

(log #c(0 1) #c(0 -1))

→
#C(-1.0 0.0)
or #C(-1 0)

(log 8.0 2)

→
3.0

(log #c(-16 16) #c(2 2))

→
3
or #c(3.0 0.0) ; approximately
or 3.0 ; approximately, unlikely

```

### Affected By

The implementation.

### Exceptional Situations

None.

• {\secref\FloatSubstitutability}

None.

\issue{REAL-NUMBER-TYPE:X3J13-MAR-89} \issue{IEEE-ATAN-BRANCH-CUT:SPLIT} \issue{IEEE-ATAN-BRANCH-CUT:SPLIT} \issue{COMPLEX-RATIONAL-RESULT:EXTEND}

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