IMLOG in Google Sheets: Choose a Base for a Complex Logarithm

• Introduced: first introduction date is not established by the reviewed official history; no desktop release number is inferred.
• Availability: documented in the rolling Google Sheets service; these examples use the web worksheet workflow.
• Checked: October 14, 2026; official IMLOG reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMLOG(value, base)

Supply both arguments. Google permits a plain numeric value as a real input and quoted complex text for both coordinates. It describes base as a positive real number; the dedicated page does not provide an omitted-base default.

Mathematically, base one is undefined because LN(1)=0. Although the source only says positive, do not treat that incomplete phrase as a guarantee that one is valid. This guide excludes base one and leaves its native error code unverified.

IMLOG is a two-argument identity. The one-argument IMLOG10 and IMLOG2 are its documented special cases when base is ten or two; they do not accept an arbitrary second base argument.

Enter 3+4i as Plain text in A2:A5. Put the numeric bases 2, 5, 10 and 0.5 in B2:B5. In C2 enter =IMLOG(A2,B2) and fill down. Extract with IMREAL and IMAGINARY into D and E.

These are rounded independent expectations. The fourth base lies between zero and one, giving a negative natural logarithm and reversing both coefficients. It is not enough to change only the real part of a result.

The input magnitude is five. That explains the real coefficient one when base is five, even though the original input is not the real number five.

For the same rows, compare:

=IMDIV(IMLN(A2),COMPLEX(LN(B2),0))

The independently expected quantities match C. IMLN supplies a complete natural logarithm, and the real divisor LN(base) scales both output coordinates.

All new complex inputs in this guide lie in the first quadrant. This avoids inventing a global negative-axis branch interval from the sparse IMLOG reference. The dedicated page establishes the special-base relationships, but not a complete branch and error table.

For base EXP(1), the natural-log divisor is mathematically one, so IMLOG agrees with IMLN as Google documents.

For each row, multiply C2 by a complex representation of LN(B2), then exponentiate:

=IMEXP(IMPRODUCT(C2,COMPLEX(LN(B2),0)))

The independent target is 3+4i for every listed base. Check real coefficient three and imaginary coefficient four separately, with a declared tolerance such as 1E-9 for this moderate fixture.

IMEXP completes the reconstruction. Retain the unrounded C result; copying the nine-decimal table values into a later formula introduces avoidable rounding differences.

A display-string comparison is less useful than a coefficient comparison because equivalent quantities can have different shortened representations.

Keep base five and compare =IMLOG("3+4i",5) with =IMLOG("15+20i",5). The second input is five times the first, with the same direction.

The second result has independently expected coefficients 2 and 0.576160914. Subtracting the first logarithm gives real one and imaginary zero.

By contrast, changing the base from five to two while keeping the original input changes both coefficients. Input scaling and base selection are different edits to a model.

Google documents =IMLOG(100,10) returning two. For positive real values, its notes explicitly relate IMLOG to ordinary LOG. A similar independent control, =IMLOG(8,2), has mathematical target three.

For a validated numeric base in B2, =AND(B2>0,B2<>1) expresses the mathematical admission rule used here. It is a rule for this worksheet model, not a complete parser for arbitrary text or errors.

The source’s =IMLOG("1+i",3.5) control has coefficients approximately 0.276647377832556 and 0.626932774314643, consistent with independent base change. Zero inputs, negative-real branch endpoints, malformed representations and exact native invalid-base errors remain unverified by this guide.

• Google IMLOG reference.
• Editorial testing standards.

Examples use fictional data, English function names and comma separators. Expected outputs are independent mathematical calculations or explicitly documented rules, not native Sheets captures. Check the spreadsheet locale when reproducing punctuation-sensitive formulas.