Logarithms Intro
What Is a Logarithm?
A logarithm tells you How Many Times you must multiply a number by itself to get another number.
Think of it as the opposite of exponentiation.
If (2^3 = 8) then the logarithm says “3 Is The Power That Makes 2 Become 8.”
We write this as
[ \log_2 8 = 3 ]
The small number at the bottom (the Base) is the number we keep multiplying. The number on the right is the result we want.
How to Read a Log
The notation (\log_b a) reads as “log base B of A.”
- Base (B) – the number we multiply.
- Argument (A) – the number we want to reach.
So (\log_10 100 = 2) means “10 multiplied by itself 2 times equals 100.”
If the base is 10, we often just write (\log a) because base‑10 logs are common in everyday life (like the Richter scale for earthquakes).
Simple Examples
| Expression | Meaning | Answer |
|---|---|---|
| (\log_3 27) | “What power of 3 gives 27?” | 3 (because (3^3 = 27)) |
| (\log_5 1) | “What power of 5 gives 1?” | 0 (any number to the power 0 is 1) |
| (\log_2 0.5) | “What power of 2 gives 0.5?” | –1 (because (2^-1 = 0.5)) |
Why Logarithms Matter
- They turn big multiplications into easier addition.
- Scientists use them to measure sound, earthquakes, and light.
- In computers, logs help with algorithms that sort and search quickly.
Now you know the basics: a logarithm answers “how many times do we multiply the base to reach the target?” Try a few on your own and see the pattern appear!