Concepts · 3 min read
Logarithms Explained Simply: What They Are and Why They Matter
The idea
If bʸ = x then log_b(x) = y. Logarithms undo exponents, just as division undoes multiplication.
| x | log₁₀ x | log₂ x |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.301 | 1 |
| 4 | 0.6021 | 2 |
| 8 | 0.9031 | 3 |
| 10 | 1 | 3.3219 |
| 100 | 2 | 6.6439 |
| 1000 | 3 | 9.9658 |
The rules
- log(xy) = log x + log y — multiplication becomes addition
- log(x/y) = log x − log y
- log(xʸ) = y · log x
- log_b x = ln x / ln b (change of base)
log₅(30) by change of base
- Natural log ln(30) = 3.401197
- Log base 10 = 1.477121
Step-by-step working 3 steps
- 1Meaning
log_5(30) = y means 5^y = 30
A logarithm answers: “what power of the base gives x?”
- 2Change of base
y = ln(30) / ln(5) = 3.40119738 / 1.60943791 = 2.11328275
- 3Check
5^2.11328275 ≈ 30
Raising the base to the answer returns x (to rounding).
Why they appear in real life
pH = −log₁₀[H⁺]: a solution with [H⁺] = 10⁻⁷ has pH 7. On the Richter scale each whole step is a tenfold increase in measured amplitude, so two steps is 100 times; one and a half steps is 31.62 times.
Practice
Try the logarithm calculator and the exponent calculator; see exponents and powers explained for the inverse side.
Frequently asked questions
What is the difference between log and ln?
log usually means base 10; ln is the natural logarithm, base e ≈ 2.718.
Why can’t I take the log of a negative number?
No real power of a positive base gives a negative result.
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