Maths: How Change-of-Base Finds a Range

Logarithms let us measure numbers by their exponents, but what happens when the base isn’t convenient? The change-of-base law is the bridge that lets you rewrite any logarithm in terms of a base you already understand—like base 2. Instead of memorising a new table for every possible base, you can express log_b 8 as log_2 8 divided by log_2 b. Since log_2 8 equals 3, this collapses to the simple relationship log_b 8 = 3 / log_2 b. This formula is more than a shortcut—it reveals how logarithms behave as the base changes. Because log_2 b grows as b increases, the fraction 3 / log_2 b shrinks, meaning larger bases produce smaller log values for the same number. That inverse relationship lets you predict not just specific values, but entire ranges: for example, knowing when log_b 8 exceeds 2 becomes a matter of comparing log_2 b to 1.5, which translates directly into a boundary for b. Understanding this single law turns scattered table entries into one coherent, flexible tool for solving exponential and logarithmic problems across any base.