How this instrument works
A logarithm answers one question: to what power must the base be raised to reach the given number? A computing engine that only knows one logarithm — natural log, ln — cannot answer that directly once the base stops being e, so it needs a bridge. That bridge is change of base: divide the natural log of the number by the natural log of whichever one you're actually measuring in. Route the same calculation through a completely different reference logarithm, say common log or log seven, and the ratio still lands on the identical value, because the reference cancels out of the algebra before the division ever happens.
The derivation is short enough to keep in your head. Call the unknown exponent y, so by definition the base you chose, raised to y, equals the number. Take the natural log of both sides and the exponent drops down in front: y times the natural log of that base equals the natural log of the number. Divide through and y stands alone — that last step is the entire formula, and it explains the shape: the denominator rescales the numerator, stretching or compressing it until the reading comes out in new units instead of e's own. Before electronic calculators existed, printed tables only ever listed one such number, almost always ten, and this exact division was how anyone converted that single table into a logarithm for any other choice at all.
Two boundaries fall straight out of the algebra. Because the natural log of one is zero, the logarithm of one is zero regardless of base — raising anything to the power zero gives one, so the answer is fixed before a single division runs. And because the denominator itself hits zero exactly when the base equals one, the formula refuses that input outright, matching the underlying question falling apart there: one raised to any power stays one, so no exponent can pick out a different result. A quieter payoff of the same algebra: swapping the two numbers flips the answer into its reciprocal — a rule most people memorize without ever seeing where it comes from.
- Type the number you want the logarithm of into the x (argument) field.
- Type the base you're measuring in — 2, 10, or anything else positive and not equal to 1 — into the Base, b field.
- Read log_b(x): the sheet divides ln(x) by ln(b) and reports the quotient directly, with no lookup table required.
- Change x or Base, b and log_b(x) recalculates instantly, so you can compare log_2(8), log_10(8), and log_8(8) side by side.
Worked example — log base 2 of 8
Set the argument to 8 and the base to 2. This sheet computes ln(8) ⁄ ln(2) = 2.0794415 ⁄ 0.6931472 = 3.0000000, and that answer needs no trust beyond arithmetic: 2 cubed is 8, so log base 2 of 8 has to equal 3, and the formula lands on exactly that figure.
The same division works the same way throughout. Set the argument to 100 and the base to 10 instead: ln(100) ⁄ ln(10) = 4.6051702 ⁄ 2.3025851 = 2.0000000, the ordinary common logarithm most calculators simply label log. Both results come from one division; only the reference being divided by changes between them.
Questions
Why can't a calculator compute a non-natural logarithm directly?
Most engines, this one included, implement a single logarithm — natural log — and derive every other one from it. Change of base is that derivation: divide the natural log of the number by the natural log of the base, and one built-in function ends up covering every case instead of a separate routine for each.
What's the most common mistake when applying this formula?
Flipping the fraction — putting the base's natural log on top instead of the argument's. The argument always sits in the numerator; the base always sits in the denominator. A quick check catches the slip: log₂(8) should equal 3, not one-third, so an unexpectedly small fractional answer usually means the ratio got inverted.
Does it matter which reference logarithm I divide through — natural, common, or something else?
No. Dividing by the natural log, by the common log, or by the log in any other valid reference base all produce the identical ratio, because whichever reference you pick cancels out of the algebra before the division runs and only the final quotient survives. This sheet always routes through the natural log, but any other choice would return the same figure to the decimal.
Why does the sheet reject a base equal to 1?
Because the natural log of one is zero, and the formula would then divide by zero. That matches the underlying question collapsing when the base is one: one raised to any power always equals one, so no exponent produces a different result, and there is nothing well-defined left to divide by.
How does swapping the argument and base relate to the reciprocal rule?
It is the same formula with the two numbers swapped. Change of base gives the reciprocal relationship directly: divide the natural log of the base by the natural log of the argument, and inverting that fraction lands you back on the original ratio. So the fact that log₂(8) and log₈(2) multiply to 1 is not a separate rule to memorize — it falls straight out of this one.
How were logarithms in unusual bases found before calculators existed?
Exactly this way. Printed tables and slide rules only ever tabulated one base, almost always ten, so anyone needing log seven of a number looked up the common logarithm of that number and of seven, then divided — the identical division this sheet performs automatically. The formula is centuries older than electronic computing; it is the reason a single table of common logarithms could answer questions in any base at all.