The expression ln(a + b) — the natural logarithm of a sum — often confuses students because it does not simplify in the same way as products or quotients.
Common Misconception
Many assume that:
ln(a + b) = ln(a) + ln(b)
This is false. The correct logarithmic identity applies only to multiplication:
ln(a × b) = ln(a) + ln(b)
Why Can’t We Simplify ln(a + b)?
Unlike multiplication, addition inside a logarithm has no general algebraic simplification. There is no standard rule to break apart ln(a + b) into simpler logarithmic terms.
However, in special cases (e.g., factoring), you might rewrite the sum:
ln(a + ab) = ln[a(1 + b)] = ln(a) + ln(1 + b)
Practical Tips
- Never split
ln(a + b)intoln(a) + ln(b). - If possible, factor the expression inside the log first.
- Use numerical methods or series expansions (like Taylor series) for approximations when needed.
Example
Evaluate or simplify where possible:
ln(2 + 6) = ln(8) ≈ 2.079
But:
ln(2) + ln(6) = ln(12) ≈ 2.485 ≠ ln(8)