Change of Base Formula Calculator

RESULT

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Type a number and a base, then press =.

The formula, both ways round

any base you like — worked out by change of base

Number

Base

No calculator has a key for log base 5. The change-of-base formula gets round that: divide the log of the number by the log of the base, in whatever base your calculator does have.

Examples — tap one and it solves

The change-of-base formula, step by step

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Look at any calculator and you will find two log keys: log for base 10 and ln for base e. There is no key for base 5, base 3 or base 7, and there never has been.

The way round that is one line of algebra. This change of base formula calculator applies it and shows the division rather than hiding it, so what appears on screen is the same working you would write on paper.

The Change of Base Formula

log_b x  =  ln x ÷ ln b

Divide the natural log of the number by the natural log of the base. That is the whole of it. log₅ 20 becomes ln 20 ÷ ln 5, which is 2.9957322736 ÷ 1.6094379124, which is 1.8613531161.

And the same sum done with base 10 gives exactly the same answer: log 20 ÷ log 5 is 1.3010299957 ÷ 0.6989700043, which is 1.8613531161 again. The change of base calculator shows both, side by side, because seeing them agree is what makes the idea stick.

Why the Base You Divide By Cancels

The formula looks arbitrary until you see where it comes from, and the proof is three lines.

Start from what a logarithm means: if log_b x = y, then b^y = x. Take the natural log of both sides and the exponent comes down in front: y × ln b = ln x. Divide both sides by ln b and you have y = ln x ÷ ln b.

Nothing in those three lines depended on the log being natural. Take log base 10 of both sides instead and you get y = log x ÷ log b, and the answer is unchanged. That is the real lesson of the change of base formula: the base you pass through cancels, so use whichever key your calculator actually has.

How to Use This Change of Base Formula Calculator

Type the number, type the base, press =. The answer appears with a note saying whether it is exact or rounded, and one tap opens the formula worked out both ways with every value filled in.

Under that is the step by step: why there is no key for that base, the formula itself, the division, and finally the same sum through base 10 as a check. Whole answers stay whole — log₂ 1024 is written as 10, not 10.0000000 — and a fractional base like one half is handled exactly too, so log_0.5 8 comes out as −3.

If you only want the value and not the working, the log calculator does the same arithmetic in fewer taps.

Change of Base Formula FAQ

What is the change of base formula?

log to base b of x equals ln x divided by ln b. The base you divide through can be anything, as long as it is the same on top and underneath — log base 10 works exactly as well as ln.

Why do you need the change of base formula?

Because a calculator only has two log keys: log for base 10 and ln for base e. Any other base has to be reached by dividing two logs you can actually press.

Does the change of base formula work with log instead of ln?

Yes. log x divided by log b gives the same answer as ln x divided by ln b. The base you pass through cancels out, which is why either one is fine.

How do you prove the change of base formula?

Start from b to the power y equals x. Take the natural log of both sides: y times ln b equals ln x. Divide by ln b and y is ln x over ln b — which is exactly the formula.

Two logs, one division, and the base you divide by disappears. Once that is clear the change of base formula stops being something to memorise.