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