Logarithm as the Inverse of Exponentiation

This WBCHSE Class 12 Commerce Business Mathematics and Basic Statistics chapter introduces the logarithm — a tool that answers the question exponentiation cannot ask directly: given a base and a result, what exponent produced that result?

Definition

For a base $a>0$, $a\neq1$, and a positive number $N$, the logarithm of $N$ to the base $a$, written $\log_a N$, is defined as the exponent to which $a$ must be raised to get $N$:

So asking "$\log_2 32 = ?$" is exactly the same question as asking "$2$ raised to what power gives $32$?" — and since $2^5=32$, the answer is $\log_2 32=5$.

Why $a>0$, $a\neq1$, and $N>0$

• The base $a$ must be positive and not equal to $1$: if $a=1$, then $1^x=1$ for every $x$, so "$1$ raised to what power gives $N$" has no unique answer unless $N=1$ itself (and even then, every $x$ would work) — the definition breaks down.
• $N$ must be positive: no real power of a positive base $a$ can ever produce a negative number or zero, so $\log_a N$ is undefined for $N\le0$.

Two immediate special values

From the definition directly: $\log_a a = 1$ (since $a^1=a$), and $\log_a 1 = 0$ (since $a^0=1$) — true for every valid base $a$.

The same logarithm concept taught here is a standard, well-established part of algebra across commerce-mathematics and general mathematics curricula nationally, applied in this WBCHSE Class 12 Commerce syllabus to simplify calculations and solve equations with the unknown in an exponent.