IMLN in Excel: Read Complex Logarithms as Magnitude and Angle

• Introduced: the first introduction version is not established by the reviewed official history.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2019 for Mac, Excel 2016.
• Checked: October 14, 2026; official IMLN reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMLN(inumber)

The required input is a complex value in i or j notation. Microsoft recommends COMPLEX for numeric coefficients.

The reference’s equation combines the logarithm of the square root of x squared plus y squared with an arctangent term. The abbreviated ATAN(y/x) expression needs quadrant handling outside the positive-real region and cannot divide by zero on the imaginary axis.

These worked outputs use positive real and imaginary coefficients, where that expression is unambiguous. The page does not separately establish every branch endpoint or zero-input error. Broader branch-sensitive rules must be checked rather than inferred from a simplified equation.

Enter the first column as text in A2:A4. Put =IMLN(A2) in B2 and fill down. Extract real and imaginary coefficients into C and D.

The source’s explicit 3+4i example corroborates the final row. Its magnitude is five, so the real logarithm component is LN(5), not LN(3).

The first two magnitudes are the square roots of two and five. Halving LN(2) and LN(5), respectively, produces their real logarithm coefficients.

Use =IMREAL(B2) and =IMAGINARY(B2) rather than manually splitting a displayed string. A minus sign or shortened decimal display is not a reliable field delimiter for a general complex record.

For each input, =LN(IMABS(A2)) has the same independently expected real coefficient as IMLN. IMABS supplies the magnitude before the ordinary real logarithm is taken.

For these strictly first-quadrant records, =ATAN(IMAGINARY(A2)/IMREAL(A2)) supplies the expected imaginary coefficient. This is a bounded check, not a replacement for a quadrant-aware complex angle function on unrestricted data.

For the first row, converting the imaginary coefficient with =DEGREES(IMAGINARY(B2)) gives 45 degrees. The value stored in the logarithm is still in radians; changing a displayed angle unit does not change the logarithm’s definition.

Put =IMEXP(B2) in E2 and fill down. Expected reconstructed quantities are 1+i, 2+i and 3+4i.

Check both coordinates, for example =ABS(IMREAL(E2)-IMREAL(A2))<0.000000001 and the corresponding IMAGINARY difference. Both are expected TRUE under these independently modeled, moderate inputs.

IMEXP restores magnitude through the real exponent and direction through the imaginary exponent. A real-only logarithm report cannot perform that reconstruction because it discarded the angle.

Displayed rounded coefficients are not full-precision replacements for B2. Keep the actual calculated value when testing the inverse identity.

Compare z=3+4i with 10z=30+40i. Their magnitudes are five and fifty, while their first-quadrant angles are identical.

The difference of their logarithms therefore has independently expected real coefficient LN(10), approximately 2.302585093, and imaginary coefficient zero:

=IMSUB(IMLN("30+40i"),IMLN("3+4i"))

The second logarithm’s real coefficient is LN(50), approximately 3.912023005. Scaling by a positive real factor leaves this direction unchanged. A negative factor would introduce a direction change, so the same unqualified argument would not apply.

The first-quadrant values 3+4i and 4+3i both have magnitude five and real logarithm component LN(5). Their angles differ: approximately 0.927295218 and 0.643501109 radians.

A comparison of only the real logarithm coefficients would call them equal even though their complete complex quantities differ. Retain the imaginary part whenever direction is relevant.

Likewise, the familiar real-number statement that logarithms turn every product into an unrestricted sum needs care for complex branches. A full turn can be discarded by a chosen argument representative. This guide does not certify branch-sensitive product identities outside its worked domain.

The mathematical logarithm is undefined at zero. An invalid complex string is a separate input problem. The dedicated reference does not supply exact error codes or complete native blank/coercion rules for those cases.

All calculations here use explicit valid complex inputs, first-quadrant angles and independent numerical models. Exact last digits, extreme magnitudes and unlisted client behavior remain unverified; no native Excel execution was performed.

• Microsoft IMLN reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.