Logarithm
log₁₀, ln, log₂ of any number
About This Calculator
A logarithm answers the question: to what power must a base be raised to produce a given number? The three most common logarithms are log base 10 (common log), the natural log base e (ln), and log base 2 (used in information theory and computing). Logarithms convert multiplication into addition, making them essential in science, engineering, and finance.
Formula
log_b(x) = y means b^y = x
log₁₀(x) = log(x); ln(x) = log_e(x); log₂(x) = log(x)/log(2)
Change of base: log_b(x) = log(x) / log(b)
log(a×b) = log(a) + log(b); log(a/b) = log(a) − log(b); log(a^n) = n×log(a)
Example Calculation
Find log₁₀(1000), ln(e²), and log₂(64).
- log₁₀(1000) = 3 (because 10³ = 1000)
- ln(e²) = 2 (because e² = e²)
- log₂(64) = 6 (because 2⁶ = 64)
log₁₀(1000)=3, ln(e²)=2, log₂(64)=6
Common Logarithm Values
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.693 | 1 |
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| e≈2.718 | 0.434 | 1 | 1.443 |
| 1000 | 3 | 6.908 | 9.966 |
Frequently Asked Questions
What is the difference between log and ln?
log (or log₁₀) uses base 10 and is common in engineering and pH calculations. ln uses base e ≈ 2.718 and appears naturally in calculus, growth/decay models, and physics.
Why can't I take the log of a negative number?
Logarithms are only defined for positive real numbers. Since no real power of a positive base produces a negative result, log(x) is undefined for x ≤ 0 in real numbers.
How are logarithms used in real life?
Logarithmic scales are used for earthquake magnitude (Richter), sound intensity (decibels), acidity (pH), and star brightness. They compress wide-ranging data into manageable scales.
What is log₂ used for in computing?
Log₂ measures information in bits. If you have n equally likely outcomes, you need log₂(n) bits to encode them. It also appears in algorithm analysis — binary search on n items takes log₂(n) steps.