What a Logarithm Asks
A logarithm answers a single question: what power do I raise the base to in order to get this number? log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation, and every property they have follows from that.
The Three Common Bases
- Base 10 (common log, written log) — used for decibels, pH and the Richter scale, because each unit is a tenfold change.
- Base e (natural log, written ln) — used throughout calculus, physics and finance, because e is the base at which the rate of growth equals the value itself.
- Base 2 (binary log) — used in computing and information theory, where each unit is a doubling.
Logarithm Rules Chart
| Rule | Form | Why it matters |
|---|---|---|
| Product | log(xy) = log(x) + log(y) | Turns multiplication into addition |
| Quotient | log(x/y) = log(x) − log(y) | Turns division into subtraction |
| Power | log(xn) = n log(x) | Turns exponentiation into multiplication |
| Base itself | logb(b) = 1 | Follows from b¹ = b |
| One | logb(1) = 0 | Follows from b⁰ = 1 |
| Change of base | logb(x) = ln(x) / ln(b) | Computes any base from the two built into calculators |
The first three rules are why logarithms were invented. Before electronic calculation, converting multiplication into addition through log tables and slide rules was the fastest way to compute, and it stayed that way for three centuries.
Common Values
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| 5 | 0.6990 | 1.6094 | 2.3219 |
| 10 | 1 | 2.3026 | 3.3219 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1,000 | 3 | 6.9078 | 9.9658 |
| 1,000,000 | 6 | 13.8155 | 19.9316 |
Logarithmic Scales
Several familiar measurements are logarithmic, which is why the numbers behave counterintuitively:
- Richter magnitude — each whole number is 10× the amplitude and roughly 32× the energy. A magnitude 7 releases about a thousand times the energy of a magnitude 5.
- Decibels — every 10 dB is a tenfold increase in intensity. 90 dB is not twice 45 dB; it is more than 30,000 times the intensity.
- pH — each unit is a tenfold change in hydrogen ion concentration. pH 4 is a hundred times more acidic than pH 6.
Frequently Asked Questions
Why is the log of a negative number undefined?
Because no real exponent applied to a positive base produces a negative result. Complex logarithms exist and are multivalued.
What is e?
About 2.71828, the base of natural growth. It arises as the limit of (1 + 1/n)n as n grows — the value continuously compounded interest converges to.
How do I compute a log in an unusual base?
Change of base: divide the natural log of the number by the natural log of the base. log₅(125) = ln(125)/ln(5) = 3.