IMLN in Google Sheets: Check Complex Log Magnitudes and Source Formula Errors

• 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 IMLN reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMLN(number)

Google permits a real numeric input or quoted complex text. For positive real inputs it explicitly equates IMLN with ordinary LN. COMPLEX can construct a value from numeric coefficients.

The reviewed reference has two presentation defects. One syntax sentence prints IIMLN, although the title, catalogue and every worked formula use IMLN. Its displayed equation also omits the outer natural logarithm around the magnitude.

The missing logarithm would contradict its own 1+i example: the magnitude is about 1.414213562, while the documented real output coefficient is about 0.346573590, which equals LN(SQRT(2)). This guide uses that consistent mathematical model and the worked outputs, without treating IIMLN as a separate alias.

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

The magnitudes are SQRT(8), thirteen and ten. Taking LN of each gives the real coefficients, while a first-quadrant angle calculation gives the imaginary coefficients.

The third real output is LN(10), not LN(8) and not the magnitude ten. Keeping the source beside both extracted coefficients makes these three quantities easier to distinguish.

In E2 use =LN(IMABS(A2)) and fill down. It agrees with C under the independent model. In F2 use =ATAN(IMAGINARY(A2)/IMREAL(A2)) and fill down for this strictly first-quadrant fixture.

The arctangent ratio is not a global quadrant-aware replacement. It cannot distinguish opposite quadrants by itself and divides by zero on an imaginary-axis input.

IMABS supplies the magnitude before LN is taken. IMREAL and IMAGINARY provide the numeric coordinates used in the bounded angle check.

For a direct positive-real control, =IMLN(10) is expected to return LN(10), approximately 2.302585093, as supported by Google’s stated real-input equivalence.

Put =IMEXP(B2) in G2 and fill down. Expected reconstructed quantities are 2+2i, 5+12i and 8+6i.

Inspect both differences, for example =ABS(IMREAL(G2)-IMREAL(A2))<0.000000001 and the corresponding IMAGINARY comparison. Both are expected TRUE for each independently modeled fixture.

IMEXP recovers magnitude from the real logarithm coefficient and direction from the imaginary coefficient. Reconstructing from C alone would lose the angle.

Use the unrounded B cells. The article’s nine-decimal displays are for readability and are not full-precision replacements for a calculated logarithm.

Doubling 5+12i gives 10+24i. The magnitudes become thirteen and 26, while their first-quadrant angles remain equal.

=IMSUB(IMLN("10+24i"),IMLN("5+12i")) therefore has independently expected real coefficient LN(2), approximately 0.693147181, and imaginary coefficient zero.

The logarithm of 26 is approximately 3.258096538. Its difference from LN(13), approximately 2.564949357, checks the real component independently.

This positive-real scaling example does not establish an unrestricted complex product rule across every branch. A logarithm’s chosen angle representative can affect identities involving direction changes.

Google explicitly gives =IMLN("-4.6") with real coefficient about 1.52605630349505 and imaginary coefficient about PI(). The real coefficient is LN(4.6).

That specific published example confirms a non-real logarithm for a negative-real input. It does not, by itself, state the service’s complete branch interval, signed-zero conventions or all negative-axis boundaries.

The guide retains that exact documented control while keeping its new data in the first quadrant. It does not infer a general branch policy from another spreadsheet product.

Zero has no finite mathematical logarithm. The dedicated page does not identify an exact zero-input error code, complete invalid-string rules or all Boolean/blank coercions.

The corrected magnitude equation is independently checked against all published numerical examples. It should not be read as native execution evidence or a guarantee for extreme magnitudes. No native Sheets execution was performed.

• Google IMLN 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.