IMLOG10 in Excel: Calculate Base-10 Complex Logs and Check Scaling

• 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 IMLOG10 reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMLOG10(inumber)

Inumber is required and uses complex i or j text. Microsoft recommends COMPLEX for a pair of numeric coefficients.

The reference’s equation expresses IMLOG10 as the natural complex logarithm multiplied by LOG10(e). That factor equals 1/LN(10), so both coefficients of the natural logarithm must be divided by LN(10).

Dividing only the real coefficient would corrupt the direction information. Ordinary LOG10 is a real-number function, not a documented parser for arbitrary complex-looking strings.

Enter the first column as text in A2:A4. Put =IMLOG10(A2) in B2 and fill down. Use =IMREAL(B2) and =IMAGINARY(B2) in C and D.

The last row is Microsoft’s explicit documented example, rounded for display. Its magnitude is five, and LOG10(5) is approximately 0.698970004.

For the first two rows, the magnitudes are the square roots of two and five. Their real components are half LOG10(2) and half LOG10(5), respectively. These independent identities check the magnitude contribution without parsing the complex output text.

In E2 enter =IMLN(A2) and fill down. Then calculate:

=IMREAL(E2)/LN(10)
=IMAGINARY(E2)/LN(10)

The two results agree with C2 and D2 under the independent model. IMLN supplies a natural complex logarithm, while ordinary LN(10) is a positive real scaling constant.

For 3+4i, the natural-log components are approximately 1.609437912 and 0.927295218. Dividing both by approximately 2.302585093 gives the common-log components in the table.

This also shows that IMLOG10’s imaginary coefficient is not the angle directly. Multiply it by LN(10) before interpreting it as the corresponding natural-log angle in radians for these first-quadrant controls.

Compare 3+4i and 30+40i. The second has ten times the magnitude and the same direction.

=IMSUB(IMLOG10("30+40i"),IMLOG10("3+4i"))

The independently expected difference has real coefficient one and imaginary coefficient zero. The second value’s real logarithm coefficient is approximately 1.698970004, while its imaginary coefficient remains approximately 0.402719196.

Scaling again to 300+400i adds another one to the real coefficient. A decimal order of magnitude therefore appears as an additive step in this bounded positive-scaling example.

This does not mean every arbitrary complex multiplication permits a branch-free logarithm addition rule. The chosen angle representative can matter when a product crosses a branch boundary.

For B2, construct a natural logarithm by multiplying both coefficients by LN(10), then exponentiate:

=IMEXP(COMPLEX(IMREAL(B2)*LN(10),IMAGINARY(B2)*LN(10)))

Expected reconstructed quantity: 1+i. Filling the corresponding formula down reconstructs 2+i and 3+4i.

Use separate IMREAL and IMAGINARY differences against the original coefficients. For these modest inputs, a stated absolute tolerance of 0.000000001 is a useful independent check; it is not a universal precision specification.

Keep full calculated values in the intermediate cells. Replacing them with the nine-decimal display values in the article intentionally loses information and may enlarge the recovery differences.

The quantities 3+4i and 4+3i both have magnitude five, so their real common-log coefficients agree. Their imaginary coefficients differ because their first-quadrant angles differ.

A report retaining only LOG10 of magnitude is useful when magnitude is the intended measurement, but it cannot reconstruct the complete complex observation. Label that reduced field explicitly rather than treating it as the entire logarithm.

The mathematical logarithm requires a nonzero input. The dedicated IMLOG10 page supplies the base-change relationship but does not present a complete parsing, coercion or native error table.

These examples stay in the first quadrant, with angle interpretation supported by the explicit source control and independently checked component equations. Native negative-axis branch details, zero-error code, malformed-input precedence and extreme magnitudes remain unverified. No native Excel execution was performed.

• Microsoft IMLOG10 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.