Log Calculator
Calculate logarithms in any base (log, ln, log2) with step-by-step working. Includes antilog mode and a properties reference card.
About Log Calculator
Calculate logarithms in any base with step-by-step working. Supports common log (base 10), natural log (base e), binary log (base 2), and any custom base. Includes an antilog mode to reverse the calculation and a properties reference card for studying.
What Is a Logarithm?
A logarithm answers the question: "What power must I raise the base to, to get this number?"
log_b(x) = y means b^y = x
Examples:
- log₁₀(100) = 2 because 10² = 100
- log₂(8) = 3 because 2³ = 8
- ln(e) = 1 because e¹ = e
- log₁₀(0.01) = -2 because 10⁻² = 0.01
The Three Common Bases
| Name | Notation | Base | Used In |
|---|---|---|---|
| Common log | log(x) or log₁₀(x) | 10 | pH scale, decibels, Richter scale, order of magnitude |
| Natural log | ln(x) or logₑ(x) | e ≈ 2.718 | Calculus, continuous growth/decay, physics, statistics |
| Binary log | log₂(x) | 2 | Computer science, information theory, algorithm complexity |
Logarithm Properties
These rules are essential for simplifying and solving logarithmic expressions:
| Property | Rule | Example |
|---|---|---|
| Product rule | log(ab) = log(a) + log(b) | log(6) = log(2) + log(3) = 0.301 + 0.477 = 0.778 |
| Quotient rule | log(a/b) = log(a) - log(b) | log(5) = log(10) - log(2) = 1 - 0.301 = 0.699 |
| Power rule | log(aⁿ) = n x log(a) | log(8) = log(2³) = 3 x log(2) = 3 x 0.301 = 0.903 |
| Change of base | log_b(x) = ln(x) / ln(b) | log₅(25) = ln(25) / ln(5) = 3.219 / 1.609 = 2 |
| Log of 1 | log_b(1) = 0 | Any base: b⁰ = 1, so log(1) = 0 |
| Log of the base | log_b(b) = 1 | log₁₀(10) = 1, ln(e) = 1, log₂(2) = 1 |
Where Logarithms Appear in Real Life
| Scale | Base | How It Works |
|---|---|---|
| Richter scale (earthquakes) | 10 | Each whole number = 10x more ground motion. Magnitude 6 is 10x stronger than 5. |
| Decibels (sound) | 10 | dB = 10 x log₁₀(I/I₀). Every +10 dB = 10x intensity. 80 dB is 10x louder than 70 dB. |
| pH (chemistry) | 10 | pH = -log₁₀[H⁺]. Each pH unit = 10x difference in acid concentration. pH 3 is 10x more acidic than pH 4. |
| Musical semitones | 2 | Frequency doubles every 12 semitones (one octave). A4 = 440 Hz, A5 = 880 Hz. |
| Algorithm complexity | 2 | Binary search runs in O(log₂ n) time. Doubling the data only adds one more step. |
| Compound interest | e | Time to grow = ln(target/start) / rate. To double: t = ln(2) / r ≈ 0.693 / r. |
The Antilog (Inverse Logarithm)
The antilog reverses a logarithm: if log₁₀(x) = 2, then antilog₁₀(2) = 10² = 100.
antilog_b(y) = b^y
This is useful for:
- Converting pH back to hydrogen ion concentration: [H⁺] = 10^(-pH)
- Converting decibels to intensity ratios: ratio = 10^(dB/10)
- Finding the original number from a log table result
Common Log Values to Know
| x | log₁₀(x) | ln(x) | log₂(x) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.693 | 1 |
| e (2.718) | 0.434 | 1 | 1.443 |
| 5 | 0.699 | 1.609 | 2.322 |
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| 1000 | 3 | 6.908 | 9.966 |
For solving equations involving logarithms, the equation solver handles step-by-step algebra. For the inverse operation (exponentiation), the exponent calculator computes any base raised to any power. To visualize log functions, the graphing calculator can plot y = log(x), y = ln(x), and more.
All calculations run in your browser. No data is sent to any server.
Frequently Asked Questions
What bases are supported?
You can use any positive base except 1. Quick buttons are provided for base 10 (common log), base e (natural log), and base 2 (binary log). For any other base, just type it in.
What is antilog?
Antilog is the reverse of a logarithm. If log base 10 of 100 is 2, then antilog base 10 of 2 is 100. It is the same as computing base raised to the power of the log value.
Why is log of 0 undefined?
There is no power you can raise a positive base to and get 0. For example, 10 raised to any real number is always positive. So log(0) has no answer in the real numbers.
What is the change of base formula?
To convert between bases, use log_b(x) = ln(x) / ln(b). This lets you compute a log in any base using only natural logarithms, which is how most calculators work internally.
Can I use this for natural logarithms?
Yes. Click the ln(e) quick base button or enter 2.71828 as the base. The tool shows the natural log alongside log base 10 and log base 2 for easy comparison.
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