IMLOG2 in Excel: Measure Complex Logarithms in Base Two

• 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 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMLOG2 reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMLOG2(inumber)

The required input is a complex number in i or j notation. Microsoft recommends COMPLEX for numeric coefficients and presents a base-change equation using the natural complex logarithm.

The scale factor LOG(e,2) equals 1/LN(2). Therefore both the real and imaginary coefficients of IMLN are divided by LN(2), approximately 0.693147181.

The source also mentions ordinary LOG for a number with a chosen base. That does not establish ordinary LOG as a parser for complex strings; use the dedicated complex function for the workflow here.

Enter 1+i, 2+2i and 4+4i as text in A2:A4. In B2 use =IMLOG2(A2) and fill down. Extract its coefficients in C and D.

Their magnitudes are the square roots of two, eight and 32. The base-two logarithms of those magnitudes are exactly 0.5, 1.5 and 2.5 in the mathematical model.

Every input has a 45-degree first-quadrant direction. Dividing its radian angle PI()/4 by LN(2) gives the common imaginary coefficient, approximately 1.133090035.

The imaginary coefficient is not itself 45 degrees or PI()/4 radians. Its logarithm-base scaling must be reversed before interpreting an angle.

Microsoft explicitly documents =IMLOG2("3+4i"). Expected coefficients are approximately 2.321928095 and 1.337804212.

The magnitude is five, so =LOG(5,2) checks the real coefficient. For this first-quadrant input, =ATAN(4/3)/LN(2) checks the imaginary coefficient.

These formulas provide separate evidence for size and direction. Multiplying 2.321928095 by LN(2) recovers the natural-log real coefficient near LN(5), but use the full calculated value when making a precision-sensitive comparison.

The simple ATAN ratio is only used where the real coordinate is positive. It is not a universal replacement for quadrant-aware angle handling.

Compare the first and last doubling-sequence rows:

=IMSUB(IMLOG2("4+4i"),IMLOG2("1+i"))

The independently expected result has real coefficient two and imaginary coefficient zero. The input magnitude increased by a factor of four, corresponding to two doublings.

Halving the first input to 0.5+0.5i instead gives a real coefficient of −0.5 and the same imaginary coefficient. A negative logarithm coefficient here means the magnitude is below one; it does not mean the original complex input is a negative real number.

Only positive scaling is being used in this identity. A negative or complex multiplier changes direction and can interact with the logarithm’s branch convention.

For B2, convert back to a natural logarithm and exponentiate:

=IMEXP(COMPLEX(IMREAL(B2)*LN(2),IMAGINARY(B2)*LN(2)))

Expected quantity: 1+i. The same pattern applied to B3 and B4 reconstructs 2+2i and 4+4i.

IMEXP uses the reconstructed real coefficient as a growth exponent and the imaginary coefficient as a radian angle. It needs both parts; retaining only C2 would reconstruct a positive real magnitude and lose the direction.

For moderate inputs, inspect both numeric component differences against a declared tolerance such as 0.000000001. This checks the modeled inverse without promising exact native text equality.

For 3+4i, its common-log real coefficient is approximately 0.698970004, whereas its base-two real coefficient is approximately 2.321928095.

Multiplying either pair of complex-log coefficients by the natural logarithm of its base produces the same natural-log model. IMLOG10 and IMLOG2 thus change the unit of the logarithm, not the underlying input quantity.

Do not compare only their printed numbers as though they used the same scale. Record the base in any exported derived-field name.

Zero has no finite mathematical logarithm. The dedicated page does not identify an exact zero-input error or a complete malformed-text/coercion table.

All worked logarithm outputs stay in the first quadrant or use positive scaling within it. Native negative-axis conventions, unlisted parsing forms and extreme precision remain unverified. No native Excel execution was performed.

• Microsoft IMLOG2 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.