LOG returns the exponent to which the chosen base must be raised to produce a positive input. Use base 2 for doublings, base 10 for orders of magnitude, or another positive base other than 1. Unlike LN, LOG can use a base you specify.

Official Apple syntax: LOG(pos-num, base). This guide uses the Apple Functions Help 15.4 entry, inspected on 2026-10-11. The documentation version identifies this help edition; it does not prove which Apple Numbers release first supported the function. The examples are mathematically expected results, not recorded runs from the Numbers application.

A systems analyst measures how many doublings are needed to reach an amount. Comparing LOG(x,2) with LOG(x) also demonstrates the optional base-10 default. If your goal is to compare values in a large table, first check the data type and the mathematical transformation you need. Logarithms change scale and do not simply reformat numbers; the exponential function reverses the natural-log transformation for valid inputs.

=LOG(pos-num, base)

The number must be > 0. The base must be > 0 and not equal to 1; Apple says base 1 returns an error because the underlying division is by zero. When the base is omitted, Apple defaults to 10. Apple’s official page defines these limits and any defaults: function reference. For input types and cross-table references, consult Apple’s value-type reference.

Decimal separators and locales: Apple’s English examples separate arguments with commas and use a decimal point in numbers. Apple says to use semicolons between arguments if your system uses a decimal comma. For instance, the English version uses =LOG(8,2); in a locale using a decimal comma, manually entered arguments may need =LOG(8;2). For a fractional base, =LOG(5.0625,1.5) may need both decimal commas and an argument semicolon: =LOG(5,0625;1,5). Confirm the local numeric formats in Numbers rather than copying punctuation blindly. See Apple’s official functions overview.

1. Open a new Numbers document and make one table called Logarithm Lab. Mark row 1 as a header.
2. Put Input in B1; put Base in C1 and Result in D1. Copy the numbers shown below into B2:B7 as actual numbers, not text.
3. For LOG, put the base values in C2:C5 and C7; leave C6 empty for the omitted-base example. Type each explicit formula shown in D2:D7; do not blindly fill through the blank-base row. Format the result cells to show six decimal places for comparison.
4. Compare the results to the independent expectations. Small final-digit differences are possible when software uses floating-point arithmetic; integer-looking results below are mathematical exact values for the chosen inputs, not native Numbers screenshots.

These displayed values were calculated independently using Python’s math functions, then rounded to six decimal places for display. Do not interpret them as six-decimal internal precision in Numbers. Values may display differently under General, Scientific, or region-specific number formats.

If your result is in another table: selecting the input cell while editing the formula lets Numbers insert a qualified reference, such as Logarithm Lab::B2; do not assume unqualified B2 refers to the intended table in a multi-table document. See Apple’s reference-value examples. Relative references normally adjust when filled; confirm formula references after moving or copying cells.

Doublings from a ratio

If B12 = 4 (starting units) and C12 = 64 (ending units), =LOG(C12/B12,2) returns 4: the ratio 16 equals 2 to the fourth power.

Change of base for checking

For a positive B13 and a valid C13 base, =LN(B13)/LN(C13) is mathematically equal to =LOG(B13,C13). Rounding can vary, so comparisons should use a tolerance. The denominator is 0 if the base equals 1, explaining the special restriction.

A base between zero and one

=LOG(4,0.5) has the mathematically expected result -2, because 0.5 to the power -2 equals 4. Negative logarithms are not automatically errors; only invalid inputs or bases are.

• Wrong power for the same number: LOG(8,2) is 3, but LOG(8) uses base 10 and is approximately 0.90309.
• Base equals 1: Apple explicitly identifies base 1 as an error case; use a different mathematically valid base.
• Base <= 0 or number <= 0: Both are outside Apple’s documented valid domain for real-number logarithms.
• Unexpected error from imported values: confirm the inputs are numeric and satisfy the documented domain. A string that looks like "100" is not the same as a confirmed numeric value. Apple distinguishes number values from string values in its value-type glossary.
• A formula gives a surprising output after copying: inspect its relative cell addresses and table qualification. When the formula refers to another table, Numbers uses :: separators, as described in Apple’s documentation.
• Displayed digits differ slightly from the expected table: the six-decimal columns are rounded for readability. Apple’s functions may compute using floating-point arithmetic, and even mathematical inverses can disagree in their low-order digits.
• Formula rejected in a non-English locale: verify the region’s decimal symbol and argument separator. Use Numbers’ formula editor rather than assuming pasted Excel or English syntax is universally accepted.
• Extreme inputs or app-specific errors: this guide does not assert a particular error-code spelling, overflow threshold, coercion policy or first-supported release because no native Numbers installation was available for testing.

Excel documents LOG(number,[base]) with the same mathematical purpose and default of 10. Apple’s function browser labels its arguments pos-num, base; do not paste Excel’s bracket notation literally into Numbers formulas. Microsoft documents the Excel counterpart in its official LOG reference, checked 2026-10-11. This confirms the ordinary mathematical operation, not identical handling of every empty cell, invalid string, floating-point boundary or user locale.

This batch has not executed the formulas in Google Sheets or verified Sheets-specific coercion, error strings, or platform/version behavior. Before copying a large production workbook, test representative values in the destination app and verify the imported cell types; similarly named formulas are not a guarantee of full parity.

• EXP

• Apple Numbers — function documentation
• Apple’s value-type reference
• Apple’s official functions overview
• EXP
• LN
• LOG10

Verification status: Official documentation checked; example outcomes were independently reasoned or arithmetically checked where applicable. Not executed in Apple Numbers.

• No genuine Apple Numbers screenshots are included; this guide does not use mock application images.