Library / Foundations: Maths / FMA-39
Foundations: Maths
Change of logarithm base
Converts a logarithm in any base to natural logarithms, which is what most calculators provide.
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- FMA-39
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- Foundations: Maths
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x is the number whose logarithm is wanted (x > 0) and b the base (b > 0, b ≠ 1). The change-of-base rule logb x = ln x / ln b works with any calculator that has an ln or log key. Special cases: base 10 for decibels, base 2 for bits, base e for natural logs.
Converts a logarithm in any base to natural logarithms, which is what most calculators provide.
Calculator
- Natural logarithm ln x4.605
- Logarithm to base 26.644
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.
- The result is the power the base must be raised to: 10² = 100, so log₁₀ 100 = 2.
- Multiplying two numbers adds their logarithms, which is why decibels turn power ratios into sums.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Base | 1.1 to 100 | 1 | 48.32 | 47.32 | +10% | −3.97 % |
| x Number | 1.01 to 1,000,000 | 0.004321 | 6 | 5.996 | +10% | +2.07 % |
Sweep of Base
| Base | Logarithm |
|---|---|
| 1.1 | 48.32 |
| 13.46 | 1.771 |
| 25.83 | 1.416 |
| 38.19 | 1.264 |
| 50.55 | 1.174 |
| 62.91 | 1.112 |
| 75.28 | 1.066 |
| 87.64 | 1.03 |
| 100 | 1 |
Sweep one variable
Variables and how each one contributes
Logarithm
What the equation gives you. Everything else feeds it.
Number
The number whose logarithm is taken. Increases the result slowly: each time x is multiplied by the base, the logarithm rises by exactly 1.
Base
The base of the logarithm. A larger base gives a smaller logarithm of the same number: log₂ 100 is 6.64 but log₁₀ 100 is 2.
Worked example with the values above
- Natural logs: ln 100 = 4.6052, ln 10 = 2.3026.
- Divide: 4.6052 / 2.3026.
- log10 100 = 2.
Assumptions and limits
- x is positive and the base is positive and not equal to 1.
- The same base is used throughout the working.
Common mistakes
- Dividing ln b by ln x instead of ln x by ln b.
- Confusing log (usually base 10) with ln (base e) on a calculator.
Statistics and uncertainty
A relative error in x changes the logarithm by an absolute amount: a 10 % error in x shifts log₁₀ x by only 0.04, which is why logarithmic scales hide small errors.
Where it is used
- Converting decibel and information measures on a calculator.
- Solving for a power or time in growth and decay problems.
Maths you need
- Indices: bˣ and its inverse
- Calculator: ln and log keys