Concept of Logarithm

A logarithm answers a single question: to what power must a fixed base be raised to produce a given number? Before calculators and computers, this idea — introduced by John Napier in the early seventeenth century and refined by Henry Briggs — was the practical tool that turned tedious multiplications, divisions, powers and roots of large numbers into simple additions and subtractions. That is exactly why it still opens a Business Mathematics course: it is the computational engine behind compound interest, growth rates, geometric means and index numbers. The Odisha CHSE +2 (First Year) Commerce Business Mathematics and Statistics syllabus draws on the same standard mathematical principles found in any established treatment of logarithms, applied here to commercial and statistical calculation.

Definition. If $a$ is a positive real number with $a \neq 1$, and $N$ is a positive number such that

then the exponent $x$ is called the logarithm of $N$ to the base $a$, and we write

These two statements say exactly the same thing, one written in exponential (index) form and the other in logarithmic form — learning to move freely between them is the whole foundation of the chapter:

For example, since $2^{5} = 32$, we may write $\log_{2} 32 = 5$; and since $10^{3} = 1000$, we may write $\log_{10} 1000 = 3$. In words, $\log_{2}32 = 5$ reads "the logarithm of 32 to the base 2 is 5", meaning 2 must be raised to the power 5 to give 32.

Two conditions must always hold for a logarithm to be defined: the base $a$ must be positive and not equal to 1 (a base of 1 gives $1^{x}=1$ for every $x$, which could never produce any other number), and the number $N$ must be strictly positive (no real power of a positive base is ever zero or negative). These restrictions are revisited as formal properties in §2.

Two systems of logarithms are in common use. The common (Briggsian) logarithm uses base 10 and is written simply as $\log N$ (the base 10 is understood when no base is shown) — this is the system used for all numerical computation and log-table work in this chapter (§4 onward). The natural (Napierian) logarithm uses the base $e \approx 2.71828$ and is written $\ln N$ or $\log_{e} N$; it appears in continuous growth and calculus contexts. Unless a base is explicitly stated in this chapter, $\log$ means $\log_{10}$.