Definition and algebraic properties of the natural logarithm

Algebraic properties of the logarithm

The key idea: converting a product into a sum

The fundamental property of the natural logarithm is that it transforms a multiplication into an addition. For all real numbers a > 0 and b > 0:

ln(a × b) = ln(a) + ln(b)

This relationship, known as a functional relationship, lies at the historical origin of the invention of logarithms: it made it possible to replace tedious multiplications with simple additions.

Practical implications

From this basic rule, we can derive other very useful formulas, provided that the arguments of the natural logarithm are strictly positive:

ln(1/a) = -ln(a)

ln(a/b) = ln(a) - ln(b)

ln(a^n) = n × ln(a) for any positive integer n

ln(√a) = (1/2) × ln(a)

Practical example

Let’s simplify ln(8) + ln(1/2) using these rules:

ln(8) + ln(1/2) = ln(8 × 1/2) = ln(4) = ln(2²) = 2 × ln(2)

Another example: solving ln(x) + ln(3) = ln(12) amounts to writing ln(3x) = ln(12), so 3x = 12 (provided that x > 0), which gives x = 4.

The classic pitfall

One must NEVER write ln(a + b) = ln(a) + ln(b): the logarithm transforms products into sums, not sums into products. Similarly, ln(a) × ln(b) is not equal to ln(a × b). You must also always check the domain of validity: before writing ln(a/b) = ln(a) – ln(b), you must ensure that a > 0 and b > 0; otherwise, the equality is meaningless.

Summary table