IMLN

Returns the natural logarithm of a complex number.

Syntax:

IMLN(complexnumber)

complexnumber is text representing a complex number, for example as a+bi or a+bj.

IMLN returns the natural logarithm of complexnumber as text.

Example:

IMLN("1+2i")

returns 0.80471895621705+1.10714871779409i as text.

Application:

The IMLN function calculates the natural logarithm of a complex number. This is a specialized function used in fields like electrical engineering, physics, and advanced mathematics.

A complex number is written in the form x+yi, where x is the real part and y is the imaginary part. The natural logarithm of a complex number z=x+yi is given by the formula:

$ln (z) = ln (∣ z ∣) + i ar g (z)$

where:

|z| is the magnitude (or modulus) of the complex number, calculated as $x^{2} + y^{2}$ .
arg(z) is the argument of the complex number, which is the angle in radians it makes with the positive real axis. This is calculated as $tan^{- 1} (\frac{y}{x})$ .

An application of using the IMLN function is in electrical engineering, particularly in the analysis of AC (alternating current) circuits. In AC circuit analysis, engineers use complex numbers to represent impedance (Z), which is the total opposition a circuit presents to an alternating current. The natural logarithm of impedance can be a step in more complex calculations for things like signal processing or analyzing wave propagation.

Let's imagine an engineer is analyzing a set of electrical components and has measured their impedances, which are represented as complex numbers. They need to find the natural logarithm of each impedance value to perform further analysis.

Here's how they could use the IMLN function:

	Component	Impedance (Complex Number)	Natural Logarithm (IMLN)
	A	B	C
1	Resistor-Inductor	3+4i	IMLN(B1)
2	Capacitor-Resistor	5-12i	IMLN(B2)
3	Inductor-Capacitor	0+8i	IMLN(B3)
4	Purely Resistive	-6+0i	IMLN(B4)

Explanation of the Table:

Column A: Component: A simple label for the electrical component being measured.
Column B: Impedance (Complex Number): The measured impedance of each component, formatted as a complex number string.
Column C: Natural Logarithm (IMLN): This is where the IMLN function is used.
- In cell C1, the formula IMLN(B1) calculates the natural logarithm of the complex number "3+4i". The result would be approximately 1.6094+0.9273i.
- In cell C2, the formula =IMLN(B2) calculates the natural logarithm of "5-12i". The result would be approximately 2.5649−1.1760i.
- And so on. The function handles both the real and imaginary parts of the complex number to provide a complex number as the result.