Understanding Logarithms: Basics and Rules

Logarithms: The Basics and Essential Rules

Logarithms are the inverse of exponentials - if a^x = n, then log_a n = x. Think of it as asking "what power do I need to raise 'a' to get 'n'?" When you see a log with no base written, it's base 10, and 'ln' means log base e (natural logarithm).

There are several fundamental logarithm rules you absolutely need to memorise. Log_a(1) always equals 0, log_a$a$ always equals 1, and log_a(0) is undefined. These make sense when you think about what powers give you these results.

The operational rules are where logarithms become really powerful. When multiplying inside a log, you can split it into addition: log_a(xy) = log_a$x$ + log_a$y$. Division becomes subtraction: log_a(x/y) = log_a$x$ - log_a$y$. Powers can be brought out front: log_a$x^n$ = n·log_a$x$.

Quick Tip: All these logarithm rules come directly from the laws of indices - they're just the flip side of exponential rules you already know!

The change of base formula log_a n = $log_b n$/$log_b a$ lets you convert between different bases, which is essential when using calculators that only have log and ln buttons.