IMLOG2 in Google Sheets: Track Complex Doublings with Correct Syntax

• 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 IMLOG2 reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMLOG2(value)

Google permits real numeric inputs and quoted complex values. It says IMLOG2 is equivalent to the general IMLOG function with base two.

The sample-formula list mistakenly adds a second argument, although the formal signature and all calculated examples use one. These formulas follow the one-argument contract; a base argument is not added to IMLOG2.

A base-two logarithm is also different from converting a number into binary text. The returned complex coefficients describe a logarithm, not a field of binary digits.

Enter 3+4i, 6+8i and 12+16i as Plain text in A2:A4. Put =IMLOG2(A2) in B2 and fill down. Extract real and imaginary components into C and D.

Their magnitudes are five, ten and twenty. The real components are LOG of those magnitudes to base two. Each successive magnitude doubles, producing an increase of one.

All inputs have the same first-quadrant angle. Dividing that radian angle by LN(2) gives the constant imaginary logarithm coefficient.

Use =IMREAL(B2) and =IMAGINARY(B2) rather than attempting to split a printed complex string. The table values are shortened independent expectations.

The source explicitly allows real numbers. =IMLOG2(8) therefore has independently expected result three, because two cubed equals eight. Google’s worked numeric control =IMLOG2(100) gives approximately 6.643856190.

Compare the numeric input eight with a complex value whose magnitude is eight but whose direction is not zero. A matching real logarithm coefficient would not mean the complete logarithm is identical; its imaginary coefficient would retain direction.

Google also explicitly documents 1+i returning coefficients 0.5 and approximately 1.133090035. Its magnitude SQRT(2) explains the exact half in the mathematical model.

For 3+4i, the real component can be checked with =LOG(5,2). The first-quadrant imaginary component can be checked with =ATAN(4/3)/LN(2).

For a general cell in this first-quadrant fixture, use IMLN and divide both components by LN(2):

=COMPLEX(IMREAL(IMLN(A2))/LN(2),IMAGINARY(IMLN(A2))/LN(2))

Both coefficients must be scaled. The imaginary coefficient 1.337804212 is not itself the original angle; multiplying by LN(2) recovers about 0.927295218 radians.

The simple ATAN ratio is used only with positive real and imaginary coordinates. It is not certified as a universal branch-aware angle calculation.

Subtract the first logarithm from the last with =IMSUB(B4,B2). The independent result has real coefficient two and imaginary coefficient zero.

Two logarithmic units in base two correspond to a factor of four in the original magnitude, matching twenty divided by five. They do not mean that the original magnitude increased by two.

Halving the first input to 1.5+2i instead gives real coefficient approximately 1.321928095 with the same imaginary coefficient. The difference from the first row is negative one, representing one halving.

These statements apply to positive-real scaling with unchanged direction. A negative or complex factor changes direction and requires additional branch-aware analysis.

For B2, use:

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

Expected reconstructed quantity: 3+4i. The corresponding filled formulas reconstruct 6+8i and 12+16i.

IMEXP restores the complete quantity after the natural-log scale is recovered. Verify both numeric coordinates against the original using a declared tolerance, such as 0.000000001 for these modest values.

Keeping only the real logarithm coefficient would reconstruct a positive magnitude without its original direction. Keep the full B value whenever subsequent operations need the complex quantity.

IMLOG10 of 3+4i has real coefficient approximately 0.698970004. IMLOG2 of the same input has real coefficient approximately 2.321928095. Both describe the same magnitude in different logarithmic units.

Multiplying each pair of coefficients by LN of its base produces the same independent natural-log model. This relationship is more informative than comparing the raw displayed numbers without recording their bases.

The dedicated reference does not establish complete invalid-input, zero-error or global branch rules. Its erroneous two-argument sample list is not followed. New complex outputs remain first-quadrant models; no native Sheets execution was performed.

• Google IMLOG2 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.