What is a Logarithm?
A logarithm answers the question: "To what power must a base be raised to get a given number?"
If \(a^x = y\) (exponential form), then \(\log_a y = x\) (logarithmic form).
Understanding the Relationship:
| Exponential Form | Logarithmic Form | Read as |
|---|---|---|
| \(3^2 = 9\) | \(\log_3 9 = 2\) | "log base 3 of 9 equals 2" |
| \(5^4 = 625\) | \(\log_5 625 = 4\) | "log base 5 of 625 equals 4" |
| \(10^2 = 100\) | \(\log_{10} 100 = 2\) | "log base 10 of 100 equals 2" |
Key Components:
- Base (a): The number being raised to a power (must be >0 and ≠1)
- Argument (y): The result after raising a to power x (must be >0)
- Exponent (x): The logarithm itself
Important Special Values:
| Condition | Exponential Form | Logarithmic Form |
|---|---|---|
| \(a^1 = a\) | \(a^1 = a\) | \(\log_a a = 1\) |
Why Use Logarithms?
- Logarithms turn multiplication into addition (easier!)
- They help solve exponential equations
- They are used in science (pH scale, Richter scale, decibels)