Natural logarithm
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For the ln
function in MATLAB^{®}, see log
.
ln(x
)
ln(x)
represents the natural logarithm of x
.
Natural logarithm is defined for all complex arguments x ≠ 0.
ln
applies the following simplification rules
to its arguments:
If x
is of the type Type::Numeric
, then .
Here k
is an integer, such that the imaginary part
of the result lies in the interval .
Similar simplifications occur for .
If x
is a negative integer or a
negative rational, then ln(x)
= i π + ln( x).
If x
is an integer, then .
ln
uses the following special values:
ln(1) = 0, ln( 1) = i π, , , ln(∞) = ∞, ln( ∞) = i π + ∞.
For exact numeric and symbolic arguments, ln
typically
returns unresolved function calls.
If an argument is a floatingpoint value, ln
returns
a floatingpoint result. The imaginary part of the result takes values
in the interval .
The negative real axis is a branch cut; the imaginary part of the
result jumps when crossing the cut. On the negative real axis, the
imaginary part is π according
to ln(x)
= i π + ln( x), x <
0. See Example 3.
If an argument is a floatingpoint interval of type DOM_INTERVAL
, ln
returns
the results of type DOM_INTERVAL
, properly rounded outwards.
This implies that the result contains only real numbers. See Example 4.
Arithmetical rules such as ln(x y)
= ln(x) + ln(y) are
not valid throughout the complex plane. Use properties to mark identifiers
as real and apply functions such as expand
, combine
or simplify
to manipulate expressions involving ln
.
See Example 5.
When called with a floatingpoint argument, the function is
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
Compute the natural logarithms of these numeric and symbolic values:
ln(2), ln(3), ln(1/4), ln(1 + I), ln(x^2)
For floatingpoint arguments, ln
returns
floatingpoint results:
ln(123.4), ln(5.6 + 7.8*I), ln(1.0/10^20)
ln
applies special simplification rules to
its arguments:
ln(1), ln(1), ln(exp(5)), ln(exp(5 + 27/4*I))
diff
, float
, limit
, series
and similar functions
handle expressions involving ln
:
diff(ln(x^2), x)
float(ln(PI + I))
limit(ln(x)/x, x = infinity)
series(x*ln(sin(x)), x = 0, 10)
The negative real axis is a branch cut. The imaginary part of
the values returned by ln
jump when crossing this
cut:
ln(2.0), ln(2.0 + I/10^1000), ln(2.0  I/10^1000)
The natural logarithm of an interval is the image set of the logarithm function over the set represented by the interval:
ln(1 ... 2)
ln(1 ... 1)
This definition extends to unions of intervals:
ln(1 ... 2 union 3 ... 4)
expand
, combine
, and simplify
react to properties
set via assume
.
The following call does not produce an expanded result, because the
arithmetical rule ln(x y)
= ln(x) + ln(y) does
not hold for arbitrary complex x, y:
expand(ln(x*y))
If one of the factors is real and positive, the rule is valid:
assume(x > 0): expand(ln(x*y))
combine(%, ln)
simplify(ln(x^3*y)  ln(x))
For further computations, clear the assumption:
unassume(x):

Arithmetical expression
x