What is a Logarithm?
A logarithm is the inverse operation of exponentiation. While exponentiation asks "what do I get when I raise base b to power k?", a logarithm asks "to what power must I raise base b to obtain x?"
Formally: logb(x) = k means bk = x
For example, log3(9) = 2 because 3² = 9. But what about log3(16)? Since 16 is not a perfect power of 3, the answer is not an integer — this is exactly where the change of base formula becomes essential.
What is the Change of Base Formula?
The change of base formula lets you rewrite any logarithm in terms of logarithms with a different (more convenient) base:
logb(x) = loga(x) / loga(b)
Here:
- x — the argument (the number inside the logarithm, must be positive)
- b — the original base (must be positive and ≠ 1)
- a — the new (intermediate) base used for computation (any valid base: 10, e, 2, etc.)
The formula works because all three expressions equal the same value regardless of which intermediate base a you choose. Scientific calculators typically have only log₁₀ and ln (natural log) buttons, so the change of base formula bridges the gap to any other base.
Example: How to Change the Base of log
Let's compute log3(16) step by step using base 10:
- Apply the formula: log3(16) = log10(16) / log10(3)
- Calculate: log10(16) ≈ 1.20412
- Calculate: log10(3) ≈ 0.47712
- Divide: 1.20412 / 0.47712 ≈ 2.52373
- Verify: 32.52373 ≈ 16 ✓
You can verify this result with our calculator above by entering x = 16, base = 3, and selecting Common (10) as the intermediate base.
Common Logarithm Bases
| Base | Name | Notation | Common Use |
|---|---|---|---|
| 10 | Common logarithm | log(x) or log₁₀(x) | pH, Richter scale, decibels, engineering |
| e ≈ 2.71828 | Natural logarithm | ln(x) | Calculus, physics, continuous growth, finance |
| 2 | Binary logarithm | log₂(x) or lb(x) | Computer science, information theory, data storage |
US Imperial vs Metric System Context
Logarithms appear in both measurement systems:
- Metric (SI): SI prefixes follow powers of 10 (kilo = 10³, mega = 10⁶, giga = 10⁹…). The common logarithm (base 10) directly maps to the order of magnitude of any SI quantity. The Richter scale and pH scale are both base-10 logarithmic scales used in metric-based science.
- US/Imperial: Computer data storage traditionally uses base-2 prefixes (IEC binary: 1 kibibyte = 2¹⁰ bytes). The binary logarithm tells you how many bits are needed to represent a value. The US/Imperial naming scale (thousand, million, billion, trillion) also relates to log₁₀.
Financial Applications (Multiple Currencies)
The change of base formula is used in financial mathematics to calculate:
- Doubling time: How many periods does it take to double money at a fixed growth rate? Answer = logbase(2)
- Growth periods: If your investment grows by factor b each period, reaching x times your initial capital takes logb(x) periods
- Rule of 72: Approximate doubling time = 72 / interest_rate% (derived from ln(2) / ln(1 + r) ≈ 0.72 / r for small r)
Our calculator supports 18 world currencies including USD, RUB, EUR, GBP, JPY, CNY, INR, BRL, CAD, AUD, CHF and more. Select a currency to see the growth table in your preferred currency unit.
Frequently Asked Questions
Why does the change of base formula work?
If logb(x) = k, then bk = x. Taking loga of both sides: loga(bk) = loga(x), so k · loga(b) = loga(x), giving k = loga(x) / loga(b). The intermediate base a cancels out in the ratio, which is why any base gives the same result.
Can I use any base as the intermediate base?
Yes! The result is always the same regardless of which valid base you choose for a. However, using base 10 or base e (natural) is most practical because scientific calculators have these functions built in.
What are the restrictions on x and b?
The argument x must be strictly positive (x > 0). The base b must be positive and not equal to 1 (b > 0, b ≠ 1). These restrictions come from the domain of the logarithm function. Negative numbers and zero do not have real-valued logarithms.
What if x equals b?
logb(b) = 1 for any valid base b. This is because b1 = b, which satisfies the definition of the logarithm.
How is this used in music theory?
The musical scale uses logarithms! An octave doubles the frequency, so going from note A (440 Hz) to a note n semitones higher gives a frequency of 440 × 2n/12 Hz. To find how many semitones separate two frequencies f₁ and f₂, use: n = 12 × log₂(f₂/f₁) — a direct application of the change of base formula.
How is the change of base formula used in computer science?
In algorithm analysis, O(log n) complexity is base-agnostic because all logarithms differ only by a constant factor. The change of base formula reveals this: logb(n) = log₂(n) / log₂(b). For practical calculations involving data sizes, log₂ directly counts the number of bits needed.