How to Use the IMEXP, IMLN, IMLOG10, and IMLOG2 Functions in Microsoft Excel for Complex Exponentials and Logarithms

Working with exponential decay, Fourier transforms, transmission line impedances, and acoustic wave equations requires evaluating exponential and logarithmic functions over complex numbers (values consisting of real and imaginary parts, \(z = x + yi\)). In conventional mathematics, taking the logarithm of negative or complex values is undefined on the real number line, causing standard spreadsheet functions like LN or EXP to fail. Microsoft Excel provides four specialised engineering functions to manage these operations: IMEXP, IMLN, IMLOG10, and IMLOG2.

Mathematical Principles of Complex Exponentials and Logarithms

These functions follow Euler’s formula and complex analytical branch definitions:

• IMEXP: Calculates the natural exponential \(e^z\) for complex \(z = x + yi\): \[ e^{x + yi} = e^x (\cos(y) + i\sin(y)) \]
• IMLN: Computes the principal natural logarithm (base \(e\)) of a complex number: \[ \ln(z) = \ln|z| + i \arg(z) \] where \(|z| = \sqrt{x^2 + y^2}\) and \(\arg(z)\) is the phase angle in radians within \((-\pi, \pi]\).
• IMLOG10: Computes the common logarithm (base 10) of a complex number using change-of-base: \[ \log_{10}(z) = \frac{\ln(z)}{\ln(10)} \]
• IMLOG2: Computes the binary logarithm (base 2) of a complex number: \[ \log_2(z) = \frac{\ln(z)}{\ln(2)} \]

Function Syntax and Arguments

Each function takes a single argument:

=IMEXP(inumber)
=IMLN(inumber)
=IMLOG10(inumber)
=IMLOG2(inumber)

• inumber: A complex number entered as text in the format "x+yi" or "x+yj", or a reference to a cell containing a formatted complex number. Real numbers entered as plain digits are automatically treated as having an imaginary coefficient of zero (\(x + 0i\)).

Practical Examples in Microsoft Excel

1. Calculating Complex Exponentials with IMEXP

Euler’s famous identity states that \(e^{i\pi} = -1\). You can verify this in Excel:

1. In cell A2, construct the imaginary value using the COMPLEX function:

   =COMPLEX(0, PI())

   Excel returns "3.14159265358979i".
2. In cell B2, apply IMEXP:

   =IMEXP(A2)

   Excel returns "-1" (with the imaginary component vanishing to zero).

For general numbers, such as \(z = 1 + 2i\):

=IMEXP("1+2i")

Output: -1.13120438375681 + 2.47172667200482i

2. Calculating Natural Logarithms with IMLN

Unlike standard =LN(-5) which returns a #NUM! error, IMLN properly evaluates negative real numbers by mapping them onto the complex plane:

=IMLN("-5")

Output: 1.6094379124341 + 3.14159265358979i (where the imaginary part is exactly \(\pi\)).

For a full complex coordinate like "3+4i":

=IMLN("3+4i")

Because \(|3+4i| = 5\) and \(\arg(3+4i) \approx 0.9273\) radians, the function returns:

1.6094379124341 + 0.927295218001612i

3. Base-10 and Base-2 Logarithms with IMLOG10 and IMLOG2

For telecommunications, decibel scaling, and digital signal information theory:

• To find the base-10 log of "3+4i":

  =IMLOG10("3+4i")

  Output: 0.698970004336019 + 0.402719194901614i
• To find the binary logarithm (base-2) of "3+4i":

  =IMLOG2("3+4i")

  Output: 2.32192809488736 + 1.33780421245656i

Extracting Amplitude and Phase for Engineering Diagrams

To plot Nyquist diagrams or Bode plots from the outputs of logarithmic complex calculations, extract the components using companion functions:

Metric Required	Excel Formula	Application
Real Magnitude Component	=IMREAL(IMLN(A2))	Attenuation in Nepers / Log gain
Phase Shift (Angle)	=IMAGINARY(IMLN(A2))	Phase displacement in radians
Phase in Degrees	=DEGREES(IMAGINARY(IMLN(A2)))	Bode plot phase angle

Troubleshooting Common Errors

• #NUM! Error: Occurs when inumber is "0" or "0+0i" for logarithmic functions (IMLN, IMLOG10, IMLOG2), as the logarithm of zero is mathematically undefined.
• #VALUE! Error: Occurs when the complex number string is formatted incorrectly (such as containing unexpected spaces, double signs, or improper letters).