Logarithm Calculator — Calculate Log, Ln, Log₂ of Any Number
Calculate logarithms instantly — log base 10, natural log (ln), log base 2, or any custom base. Understand log rules, properties, and real-world applications.
A logarithm answers the question: "What power do I raise the base to, to get this number?" If 10² = 100, then log₁₀(100) = 2. Simple concept, endlessly useful in science, engineering, finance, and computing.
The CalcHub Logarithm Calculator computes logarithms in any base instantly.
Common Logarithm Values
Log Base 10 (Common Log)
| x | log₁₀(x) | Why |
|---|---|---|
| 1 | 0 | 10⁰ = 1 |
| 2 | 0.3010 | — |
| 3 | 0.4771 | — |
| 5 | 0.6990 | — |
| 10 | 1 | 10¹ = 10 |
| 100 | 2 | 10² = 100 |
| 1,000 | 3 | 10³ = 1,000 |
| 10,000 | 4 | 10⁴ = 10,000 |
| 0.1 | −1 | 10⁻¹ = 0.1 |
| 0.01 | −2 | 10⁻² = 0.01 |
Natural Log (ln, Base e ≈ 2.718)
| x | ln(x) | x | ln(x) |
|---|---|---|---|
| 1 | 0 | 10 | 2.3026 |
| 2 | 0.6931 | 20 | 2.9957 |
| 3 | 1.0986 | 50 | 3.9120 |
| e (2.718) | 1 | 100 | 4.6052 |
| 5 | 1.6094 | 1,000 | 6.9078 |
Log Base 2 (Binary Log)
| x | log₂(x) | Context |
|---|---|---|
| 1 | 0 | 2⁰ = 1 |
| 2 | 1 | 2¹ = 2 |
| 4 | 2 | 2² = 4 |
| 8 | 3 | 2³ = 8 |
| 16 | 4 | 2⁴ = 16 |
| 32 | 5 | 2⁵ = 32 |
| 64 | 6 | 2⁶ = 64 |
| 128 | 7 | 2⁷ = 128 |
| 256 | 8 | 1 byte (8 bits) |
| 1,024 | 10 | 1 KB |
| 1,048,576 | 20 | 1 MB |
Log Properties (Rules)
| Rule | Formula | Example |
|---|---|---|
| Product rule | log(a × b) = log(a) + log(b) | log(20) = log(4) + log(5) |
| Quotient rule | log(a / b) = log(a) − log(b) | log(25) = log(100) − log(4) |
| Power rule | log(aⁿ) = n × log(a) | log(1000) = 3 × log(10) = 3 |
| Change of base | log_b(x) = log(x) / log(b) | log₂(8) = log(8)/log(2) = 3 |
| Log of 1 | log(1) = 0 | Any base |
| Log of base | log_b(b) = 1 | log₁₀(10) = 1, ln(e) = 1 |
| Inverse | b^(log_b(x)) = x | 10^(log₁₀(100)) = 100 |
Where Logarithms Are Used
Earthquakes (Richter Scale)
Each whole number on the Richter scale = 10× more amplitude. A magnitude 7 earthquake is 10× stronger than magnitude 6 and 100× stronger than magnitude 5. This is a log₁₀ scale.Sound (Decibels)
dB = 10 × log₁₀(P₁/P₀)| Sound | Decibels | Times Louder Than Threshold |
|---|---|---|
| Whisper | 30 dB | 1,000× |
| Normal conversation | 60 dB | 1,000,000× |
| Rock concert | 110 dB | 100,000,000,000× |
pH (Chemistry)
pH = −log₁₀[H⁺]pH 7 = neutral, pH 3 = 10,000× more acidic than pH 7.
Computing (Algorithm Complexity)
Binary search has O(log₂ n) complexity. Searching 1 billion items takes only log₂(1,000,000,000) ≈ 30 steps.Finance (Rule of 72)
Doubling time ≈ 72 / interest rate (which comes from ln(2)/r ≈ 0.693/r)At 8% interest, money doubles in ~9 years.
Information Theory (Bits)
The information content of an event with probability p is −log₂(p) bits. A coin flip (p=0.5) contains 1 bit of information.How to Use the Calculator
- Open the CalcHub Logarithm Calculator
- Enter the number (x)
- Select the base (10, e, 2, or custom)
- See: logarithm value, with step-by-step explanation
Can you take the log of a negative number?
Not with real numbers. log(−5) is undefined in real number math. In complex analysis, it's defined using Euler's formula, but that's advanced mathematics. For practical purposes: log requires a positive input.
What's the difference between log and ln?
"log" usually means log base 10 (common log). "ln" means log base e ≈ 2.718 (natural log). In some fields (especially pure math and computer science), "log" means ln. Context matters — always check which base is intended.
Why is ln used more than log in calculus?
Because the derivative of ln(x) is simply 1/x, and the derivative of eˣ is eˣ — the cleanest possible formulas. Base 10 doesn't have this elegant property. Natural log is "natural" because it makes calculus simpler.
Related Calculators
- Exponent Calculator — inverse of logarithms
- Scientific Calculator — full function math
- Square Root Calculator — another common operation
- Compound Interest Calculator — uses ln for continuous compounding