Logarithm Formula Explained, With Examples
Every logarithm result comes from one formula. Once you understand what goes into it, you can sanity-check any figure you are given, whether by an employer, a bank, a teacher or another website.
The formula
log_b(x) = ln(x) ÷ ln(b).
What each input means
- Number: the value you enter in the calculator.
- Base: the value you enter in the calculator.
Example
For example, with these inputs:
- Number: 1,000
- Base: 10
the calculator returns:
- log_b(x): 3
- Natural log ln(x): 6.9078
How the result changes
The table keeps the other inputs at their example values and changes number.
| Number | log_b(x) | Natural log ln(x) |
|---|---|---|
| 1,000 | 3 | 6.9078 |
| 500 | 2.699 | 6.2146 |
| 750 | 2.8751 | 6.6201 |
| 1,250 | 3.0969 | 7.1309 |
| 1,500 | 3.1761 | 7.3132 |
Why it matters
These calculations come up in school, at work and in daily life, from PSLE and O-Level practice to measuring a room for new flooring. Doing them by hand builds understanding, and a calculator lets you check the answer instantly.
Each formula has assumptions built in. Area formulas assume flat shapes, volume formulas assume regular solids and statistical measures assume the data you enter is the data you want to describe. Knowing the assumptions helps you pick the right tool.
A quick mental estimate before calculating is a powerful check. If a room's floor area comes out at 4,000 square metres or a percentage comes out above 100% when it should not, you know to look again.
In everyday life the same maths shows up in discounts, recipe scaling, renovation measurements and comparing prices per unit. Being comfortable with it saves money and avoids costly measuring mistakes.
Related calculations
- Area of an Ellipse Calculator: Calculates the area of an ellipse from the measurements you enter.
- Area of a Rhombus Calculator: Calculates the area of a rhombus from the measurements you enter.
- Area of a Sector Calculator: Calculates the area of a sector from the measurements you enter.
- Circumference of a Circle Calculator: Calculates the circumference of a circle from the measurements you enter.
Use the Logarithm Calculator to plug in your own numbers.
Run your own numbers: Logarithm Calculator