The natural logarithm

Summary

The natural logarithm

  • For any \(c>0\) the equation

\[e^a=c\]

  • where \(e=2.71828…\) has a unique solution \(a\)
  • The solution \(a\) is denoted by \(ln c\) and pronounced the natural logarithm of \(c\) , the logarithm to base \(e\) , or simply the logarithm of \(c\) if it is understood that it is the natural one.
  • If \(c≤0\) then \(ln c\) is not defined.
  • \(ln c\) is sometimes denoted \(log c\) . \(log c\) may mean either the common logarithm or the natural logarithm – you need to check how it is defined.

Logarithmic identities

  • \(ln e^a=a\) for all \(a\)
  • \(e^{ln c}=c\) for \(c>0\)

Logarithm of products, ratios and powers

  • \(ln cd=ln c+ln d\)
  • \(ln \frac{c}{d}=ln c-ln d\)
  • \(ln c^d=dln c\)