Library / Foundations: Maths / FMA-21
Foundations: Maths
Logarithms
The power x such that aˣ = y.
- Ref
- FMA-21
- Rev
- v1
- Field
- Foundations: Maths
- Result
- x
a > 0, a ≠ 1 and y > 0. Logarithms turn products into sums and powers into multipliers. The calculator uses x = ln y / ln a.
A logarithm answers the question: what power must the base be raised to in order to give this number?
Calculator
- log₁₀ y0.9031
- Natural log ln y2.079
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Multiplying numbers adds their logarithms, which is why decibels turn power ratios into simple sums.
- Doubling time and half-life problems reduce to a logarithm.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| y Number y | 2 to 1,000 | 1 | 9.966 | 8.966 | +10% | +4.58 % |
| a Base a | 1.5 to 10 | 0.9031 | 5.129 | 4.225 | +10% | −12.1 % |
Sweep of Number y
| Number y | Logarithm loga y |
|---|---|
| 2 | 1 |
| 4.349 | 2.121 |
| 9.457 | 3.241 |
| 20.57 | 4.362 |
| 44.72 | 5.483 |
| 97.25 | 6.604 |
| 211.5 | 7.724 |
| 459.9 | 8.845 |
| 1,000 | 9.966 |
Sweep one variable
Variables and how each one contributes
Logarithm loga y
What the equation gives you. Everything else feeds it.
Base a
The base. A larger base gives a smaller logarithm of the same number.
Number y
The number whose logarithm is wanted. Increases slowly: each doubling of y adds a fixed amount.
Worked example with the values above
- Take natural logs: ln 8 = 2.0794, ln 2 = 0.69315.
- Divide: 2.0794 / 0.69315.
- x = 3.
Assumptions and limits
- y > 0 and a > 0, a ≠ 1.
- Same base on both sides of the equation.
Common mistakes
- Taking the log of zero or a negative number.
- Mixing up the base with the number.
Statistics and uncertainty
Because log compresses ratios, a 10% error in y changes log₁₀ y by only 0.04, which is why logarithmic scales hide small errors.
Where it is used
- Decibel and signal calculations.
- Growth, decay and time-to-threshold problems.
Maths you need
- Indices: aˣ
- Calculator: ln and log buttons