What is a Logarithm?
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.
Fundamental Identity
logb(x) = y ⟺ by = x
Where b is the base (b > 0, b ≠ 1) and x is the argument (x > 0).
Common vs. Natural Logarithms
While a logarithm can be calculated for any positive base, two specific bases are used almost universally in science and engineering:
Common Logarithm (log10)
Used in base-10 number systems. Common in acoustics (decibels), chemistry (pH), and geology (Richter scale).
Natural Logarithm (ln)
Uses the mathematical constant e (≈ 2.71828) as the base. Vital for calculus, physics, and financial modeling.
Common Logarithmic Values
| Number (x) | Common Log (log10) | Natural Log (ln) | Binary Log (log2) |
|---|---|---|---|
| 1 | 0.0000 | 0.0000 | 0.0000 |
| 2 | 0.3010 | 0.6931 | 1.0000 |
| 5 | 0.6990 | 1.6094 | 2.3219 |
| 10 | 1.0000 | 2.3026 | 3.3219 |
| 20 | 1.3010 | 2.9957 | 4.3219 |
| 50 | 1.6990 | 3.9120 | 5.6439 |
| 100 | 2.0000 | 4.6052 | 6.6439 |