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Before calculators, an astronomer multiplying two seven-digit numbers could spend minutes on one product and make a slip anywhere. John Napier, a Scottish landowner who did mathematics in his spare time, built a tool that turned multiplication into addition. Logarithms are now in the JEE Main maths syllabus, and the reason they were invented explains why the rules work. This post covers Napier's life, his idea, how Henry Briggs changed it, and worked examples.
Who was John Napier?
The MacTutor biography records that he was born in 1550 at Merchiston Castle, Edinburgh, and died on 4 April 1617. He entered St Andrews University in 1563 at the age of 13 but left without a degree, probably to study in mainland Europe. Back in Scotland he ran the family estates, experimented with improving farmland using salts, and wrote a book on the Book of Revelation (1593) that he considered his most important work.
Mathematics was a hobby for him. He wrote that he often struggled to find time for calculations between his theological work. Much of the work on logarithms was done at Gartness, where, a local account says, he would sometimes ask the miller to stop the mill so its clatter did not break his train of thought.
What problem was Napier trying to solve?
He published his tables in 1614 as Mirifici logarithmorum canonis descriptio (A description of the wonderful canon of logarithms). An English translation by Edward Wright followed in 1616. In the preface, as MacTutor quotes from that translation, Napier says that nothing troubles mathematical practice more than multiplying, dividing and taking square and cube roots of large numbers. These take a long time and invite errors. His method replaces the numbers with others that do the same job using only addition, subtraction and division by two or three.
The key fact is the rule you know today: log(ab) = log a + log b. Look up two logarithms, add them, then look up which number has that logarithm.
How were Napier's logarithms different from ours?
Napier did not think of logarithms algebraically, since algebra was not yet developed enough. MacTutor explains that he used a picture of two moving points. One moves along a line at constant speed. The other moves along a line of fixed length, with a speed equal to its distance from the end. The distance covered by the first point is the logarithm of the distance left for the second.
He chose that fixed length to be 10⁷, because the most accurate tables of sines then available had seven places. One awkward result was that, in his system, the logarithm of 1 is not 0.
How did logarithms become base 10?
The English mathematician Henry Briggs read the 1614 book and travelled to Edinburgh to meet Napier in summer 1615. MacTutor records that Briggs had already suggested logarithms to base 10, and Napier said he had the same idea but was too ill to make new tables. They agreed on base 10 with log 1 = 0. Briggs spent a month with Napier, visited again in 1616, and planned a third visit, but Napier died that spring. Briggs went on to publish base-10 tables, including his Arithmetica logarithmica of 1624, which gives the logarithms of the numbers from 1 to 20,000 and from 90,000 to 100,000.
In 1617 Napier also published Rabdologiae, describing numbered rods for multiplying, made of ivory and so known as Napier's bones. MacTutor notes that Kepler used logarithms to reduce his observations, the work that later supported Newton's theory of gravitation.
Worked example 1: multiply with logarithms
Find 347 × 28.6, using log₁₀ values to four decimal places.
log 347 = 2.5403 (since 347 = 3.47 × 10², the 2 is the power of ten).
log 28.6 = 1.4564.
Add: 2.5403 + 1.4564 = 3.9967.
Antilog: 10^3.9967 = 10^0.9967 × 10³ ≈ 9.924 × 10³ = 9924.
Direct multiplication gives 9924.2, so four-place logarithms are good to about four figures.
Worked example 2: how many digits does 2⁵⁰ have?
log 2⁵⁰ = 50 × log 2 = 50 × 0.30103 = 15.05.
A number whose logarithm is between 15 and 16 lies between 10¹⁵ and 10¹⁶, so it has 16 digits.
The answer is 16; 2⁵⁰ = 1,125,899,906,842,624.
Mistakes to avoid:
Treating log(a + b) as log a + log b. The rule is for products, not sums.
Losing the power of ten. Write each number as a value between 1 and 10 times a power of 10 before you start.
Counting digits from the logarithm alone. A logarithm of 15.05 means 16 digits, not 15.
Logarithmic functions appear in the official JEE (Main) 2026 syllabus; check the version for your exam year. If the laws of exponents behind logarithms are shaky, our guide to exponents and powers is the place to start, and the decimal notation Napier used came from Simon Stevin. To practise log problems with each step explained, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from how he worked?
Start from a real annoyance. Long multiplication was slow and error-prone, so he built something to replace it.
Make time for it. He fitted his calculations around his estates and his theology.
Welcome a better version. He accepted Briggs's base 10 even though it meant new tables.
Picture the idea. Two moving points made the concept concrete before the algebra existed.
Try it: use log 3 = 0.4771 to find how many digits 3⁴⁰ has.
