What is log?
A log, short for logarithm, is a math tool that tells you how many times you need to multiply one number (called the base) to get another number. For example, log₂ 8 = 3 because you multiply 2 by itself three times (2 × 2 × 2) to reach 8.
Let's break it down
- Base: the number you repeatedly multiply (common bases are 10 and 2, and the natural base e ≈ 2.718).
- Argument: the number you want to reach (the number inside the log).
- Notation: log₍base₎(argument). If the base is 10, we just write log( ) and if the base is e, we write ln( ).
- Example: log₁₀(100) = 2 because 10 × 10 = 100.
Why does it matter?
Logs turn multiplication into addition and division into subtraction, which makes complex calculations easier. They also help us understand how quickly things grow or shrink, such as sound intensity, earthquake strength, or computer algorithm performance.
Where is it used?
- Science: measuring decibels (sound), pH (acidity), and Richter scale (earthquakes).
- Finance: compound interest and exponential growth models.
- Computer science: analyzing algorithm speed (Big O notation) and data compression.
- Engineering: signal processing and control systems.
- Everyday: smartphone camera exposure settings and audio volume controls.
Good things about it
- Simplifies big numbers and exponential relationships.
- Provides a clear way to compare rates of growth.
- Widely supported in calculators, spreadsheets, and programming languages.
- Enables powerful mathematical techniques like solving equations with unknown exponents.
Not-so-good things
- Can be confusing at first because it works opposite to regular arithmetic.
- Requires remembering different bases and their special symbols (log, ln).
- Some calculators and software use different default bases, leading to mistakes.
- Not always intuitive for visualizing real‑world problems without practice.