Logarithm Calculator
Calculate log base 10, natural log (ln), log base 2, or any custom base. Also compute antilogarithms. Full decimal and scientific notation output with step-by-step working. Covers all A-Level and university logarithm types.
log₁₀ Reference Table (1 to 10)
| x | log₁₀(x) | x as power of 10 |
|---|---|---|
| 1 | 0 | 10&sup0 |
| 2 | 0.30103 | 10⁰⋅⁰3 |
| 3 | 0.47712 | 10⁰⋅⁴⁷ |
| 4 | 0.60206 | 10⁰⋅⁶⁰ |
| 5 | 0.69897 | 10⁰⋅⁶⁹ |
| 6 | 0.77815 | 10⁰⋅⁷⁸ |
| 7 | 0.84510 | 10⁰⋅⁰⁴ |
| 8 | 0.90309 | 10⁰⋅⁹⁰ |
| 9 | 0.95424 | 10⁰⋅⁹⁵ |
| 10 | 1 | 10¹ |
What is a Logarithm?
A logarithm is the inverse operation of exponentiation. If you know that 10³ = 1,000, then log₁₀(1,000) = 3. More formally: logₙ(x) = y means bⁿ = x. The logarithm answers the question: “to what power must I raise the base b to get x?”
Logarithms were invented by John Napier (Scottish mathematician) in 1614, originally to simplify astronomical calculations by transforming multiplication into addition. Before calculators, navigators and scientists used printed log tables to perform complex arithmetic.
Types of Logarithm
- Common logarithm (log): Base 10. Written log(x) or log₁₀(x). Used in the Richter scale, decibels, pH chemistry, and GCSE/A-Level maths.
- Natural logarithm (ln): Base e (Euler’s number, ≈ 2.71828). Written ln(x). Used in calculus, exponential growth and decay, and compound interest.
- Binary logarithm (log₂): Base 2. Written log₂(x). Used in computer science, information theory, and binary search algorithms.
- Custom base: Any positive number other than 1. Use the change-of-base formula.
The Three Laws of Logarithms
These laws apply to logarithms of any base and are essential knowledge for A-Level Maths:
Example: log(6) = log(2) + log(3) = 0.301 + 0.477 = 0.778
Example: log(5) = log(10) − log(2) = 1 − 0.301 = 0.699
Example: log(1000) = log(10³) = 3 × log(10) = 3 × 1 = 3
Change of Base Formula
To calculate a logarithm in any base using a standard calculator (which only has log₁₀ and ln):
Example: log₅(125) = log(125) ÷ log(5) = 2.097 ÷ 0.699 = 3
A-Level Maths: Logarithms and Exponentials
In A-Level Maths (Year 1 and 2), logarithms appear in several key topics:
- Solving exponential equations: 3ⁿ = 20 → n = log(20)/log(3) = 2.727
- Exponential growth and decay: N = N₀eǚ relates to ln
- Logarithmic graphs: y = log(x) and y = eˣ are inverses; their graphs reflect in y = x
- Disguised quadratics: equations like 2log(x) − log(x+3) = 1
A-Level Worked Example
Solve: 2ⁿ = 15
- Take log of both sides: log(2ⁿ) = log(15)
- Apply power rule: n × log(2) = log(15)
- Divide: n = log(15) ÷ log(2) = 1.17609 ÷ 0.30103 = 3.907 (3 d.p.)
Real-World Applications of Logarithms
| Application | Formula | Notes |
|---|---|---|
| Richter Scale (earthquakes) | M = log₁₀(A/A₀) | Each unit = 10× amplitude |
| Decibels (sound) | dB = 10 log₁₀(I/I₀) | 60dB is 1000× louder than 30dB |
| pH (acidity) | pH = −log₁₀[H⁺] | Each pH unit = 10× more/less acidic |
| Compound interest (continuous) | t = ln(A/P) ÷ r | Time for investment to grow |
| Radioactive decay | t₀.₅ = ln(2) ÷ λ | Half-life formula |
| Binary search | Steps = log₂(n) | Searching 1 billion items: only 30 steps |
How to Calculate log on a Scientific Calculator
On most scientific calculators (Casio fx-85GT, fx-991, Sharp, Texas Instruments):
- log₁₀(x): Press the log button, enter x, press =
- ln(x): Press the ln button, enter x, press =
- logₙ(x): Calculate log(x) ÷ log(b) using the change-of-base formula
- Antilog base 10: Use the 10ˣ or 10^ function
- Antilog (natural): Use the eˣ or exp function
How to Calculate log in Excel
=LOG10(x)— Common log (base 10)=LN(x)— Natural log (base e)=LOG(x, b)— Log of x in base b=10^y— Antilog base 10=EXP(y)— Antilog natural (eˣ)
How the Logarithm Calculator Works
This calculator uses the current UK grading system and educational standards to help students, parents, and teachers understand academic performance. UK education follows specific grading frameworks that differ between GCSEs, A-Levels, and university degrees.
Understanding how grades are calculated and what they mean for future progression is important for making informed decisions about subject choices, university applications, and career planning.
Key Information for 2025/26
GCSEs in England use the 9-1 grading scale, where 9 is the highest and 4 is a standard pass (equivalent to the old C grade). A-Levels use the A*-E scale with UCAS tariff points ranging from 56 (A*) to 16 (E). Universities typically require grades between AAA and CCC depending on the course and institution, with highly competitive courses often asking for A*A*A.
Example Calculation
A student achieving grades of A*, A, and B at A-Level would earn UCAS tariff points of 56 + 48 + 40 = 144 points total. This exceeds the typical entry requirement for most Russell Group universities (128 points or AAB equivalent) and would make the student competitive for many courses.
Source: Based on Ofqual and UCAS 2025/26 guidelines. Last updated March 2026.
Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse of exponentiation. logₙ(x) = y means bⁿ = x. It answers: “to what power must I raise b to get x?” For example, log₁₀(10,000) = 4 because 10⁴ = 10,000. Logarithms are fundamental to mathematics, science, and computing because they convert exponential relationships into linear ones, making calculation and analysis far simpler.
What is the difference between log and ln?
In UK school maths, log means log base 10 (common logarithm). ln is the natural logarithm with base e (≈ 2.71828). The key difference: log₁₀(10) = 1, while ln(e) = 1. In pure mathematics and some university texts, “log” may mean ln, so always check the context. In A-Level Maths, both log and ln are tested — log for solving exponential equations, ln for calculus and exponential growth/decay problems.
How do you calculate log without a calculator?
Use log laws and known values: log₁₀(1) = 0, log₁₀(10) = 1, log₁₀(100) = 2, etc. For example: log(500) = log(5 × 100) = log(5) + log(100) = log(5) + 2. Since log(5) = log(10/2) = log(10) − log(2) = 1 − 0.301 = 0.699, then log(500) = 0.699 + 2 = 2.699. Learning key values like log(2) = 0.301 and log(3) = 0.477 allows you to build many other log values by hand.
What are the laws of logarithms?
The three core laws: (1) Product Rule: log(AB) = log(A) + log(B). (2) Quotient Rule: log(A/B) = log(A) − log(B). (3) Power Rule: log(Aⁿ) = n × log(A). Additional identities: logₙ(b) = 1; logₙ(1) = 0; logₙ(bˣ) = x; b^(logₙ(x)) = x. These laws hold for any valid base and are tested in A-Level Maths (Edexcel, AQA, OCR) and university mathematics.
What is antilog?
The antilog (antilogarithm) is the inverse of a logarithm. If logₙ(x) = y, then antilogₙ(y) = bⁿ = x. For base 10: antilog(3) = 10³ = 1,000.
For natural log: antiln(1) = e¹ = e ≈ 2.718. On a scientific calculator, antilog base 10 is the 10ˣ button; antilog (natural) is the eˣ button. In Excel: =10^y for base 10 antilog; =EXP(y) for natural antilog.
How is log used in real life?
Logarithms appear in many fields: the Richter scale (earthquake magnitude) is base-10 log, so a magnitude 7 earthquake is 10 times more powerful than magnitude 6. Decibels (dB) measure sound intensity logarithmically — normal conversation (~60dB) is 1,000 times louder than a whisper (~30dB). The pH scale is −log[H⁺], so pH 4 is 10 times more acidic than pH 5. Binary search in computer science takes log₂(n) steps, meaning searching 1 billion items takes only 30 comparisons.