What are Logarithms? Explained with Examples
Logarithms often look intimidating, but they are simply a different way of writing exponents. Once you see a logarithm as the inverse of raising a number to a power, the topic becomes much clearer. This guide explains logarithms with worked examples and the essential log rules every CBSE, ICSE and JEE Maths student needs.
Logarithms are the inverse of exponents
A logarithm answers the question: "To what power must I raise the base to get this number?" In symbols, if bˣ = n, then logb n = x.
For example, since 2³ = 8, we can write log2 8 = 3. The logarithm (3) is just the exponent. Here are a few more:
- log10 100 = 2 because 10² = 100.
- log5 25 = 2 because 5² = 25.
- log3 1 = 0 because any base raised to the power 0 equals 1.
The three main log rules
Logarithms turn multiplication into addition, which is why they were historically used to simplify hard calculations. The three core rules are:
- Product rule: log(m × n) = log m + log n. Example: log2(4 × 8) = log2 4 + log2 8 = 2 + 3 = 5, and indeed 4 × 8 = 32 = 2⁵.
- Quotient rule: log(m ÷ n) = log m − log n. Example: log2(16 ÷ 2) = 4 − 1 = 3, and 16 ÷ 2 = 8 = 2³.
- Power rule: log(mᵏ) = k × log m. Example: log10(10³) = 3 × log10 10 = 3 × 1 = 3.
Common log vs natural log
Two bases appear most often:
- Common logarithm: base 10, written as log x or log10 x. It is widely used in science, including the pH scale and the Richter scale for earthquakes.
- Natural logarithm: base e (about 2.718), written as ln x. It appears throughout calculus, growth and decay problems.
For example, log 1000 = 3 (base 10), while ln e = 1 because e¹ = e.
A worked example
Suppose we want to simplify log2 32 + log2 2. Using the product rule, this equals log2(32 × 2) = log2 64 = 6, since 2⁶ = 64. Notice how the rules let us combine logs quickly without a calculator.
Why logarithms matter
Logarithms are essential for solving exponential equations, modelling population growth, and understanding many science formulas. They are a fixed part of the Class 11 syllabus and feed directly into calculus in Class 12 Maths. Because the rules are easy to mix up, it helps to practise them using our tips on how to memorise maths formulas. Master the inverse relationship first, then drill the three rules until they feel automatic.
Frequently asked questions
What is a logarithm in simple words?
A logarithm is the power to which a base must be raised to produce a given number. For example, log base 2 of 8 is 3 because 2 raised to the power 3 equals 8. It is simply the inverse of an exponent.
What is the difference between log and ln?
"Log" usually means the common logarithm with base 10, while "ln" means the natural logarithm with base e, approximately 2.718. Natural logs are used heavily in calculus, while common logs appear often in science and engineering.
Why is log of 1 always zero?
The logarithm of 1 is always 0, regardless of the base, because any non-zero number raised to the power 0 equals 1. So log base b of 1 equals 0 for every valid base b.