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ln (MuPAD)

Natural logarithm

For the ln function in MATLAB®, see log.

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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 floating-point value, ln returns a floating-point 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 floating-point 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(xy) = 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.

Environment Interactions

When called with a floating-point argument, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

Compute the natural logarithms of these numeric and symbolic values:

ln(2), ln(-3), ln(1/4), ln(1 + I), ln(x^2)

For floating-point arguments, ln returns floating-point 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))

Example 2

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)

Example 3

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)

Example 4

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)

Example 5

expand, combine, and simplify react to properties set via assume. The following call does not produce an expanded result, because the arithmetical rule ln(xy) = ln(x) + ln(y) does not hold for arbitrary complex 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:




An arithmetical expression

Return Values

Arithmetical expression

Overloaded By


See Also

MuPAD Functions

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