Logarithms answer “what power gives this number?” LN is the natural log (base e), LOG10 is base 10, and LOG lets you pick any base — for growth rates, decibels, and pH.
The example
LOG of 1000 base 10 = 3.
| A | B | |
|---|---|---|
| 1 | Expression | Result |
| 2 | LOG10(1000) | 3 |
| 3 | LN(e) | 1 |
The formula
The formula:
How it works
How it works:
LN(value)is the natural log (base e) — used in continuous growth and decay.LOG10(value)is base 10 — orders of magnitude, decibels, pH.LOG(value, base)takes any base.- Logs turn multiplication into addition — the reason they power growth-rate and compounding math (e.g. doubling time =
LN(2)/LN(1+rate)).
Change of base: any log can be written with another — LOG(x, b) = LN(x)/LN(b). And the inverse of LN is EXP; of LOG10 is 10^x; of LOG(,b) is b^x.
Try it: interactive demo
Value and base.
Variations
Natural log
Base e:
Base 10
Magnitude:
Doubling time
Logs in action:
Pitfalls & errors
Positive inputs only. Logs of zero or negative numbers error.
Base ≠ 1. LOG with base 1 is undefined.
LN vs LOG. LN is base e; LOG defaults to base 10 if you omit the base.
Practice workbook
Frequently asked questions
How do I calculate a logarithm in Excel?
How do I change the base of a log?
Why does LOG return an error?
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