What is the value of ln \(e^{4} ?\)
Answer 1:
In \(e^{4}=4\)
Explanation:
Remember
\(\ln a=b\)
means
\(e^{b}=a\)
So if \(\ln e^{4}=b\)
then
\(e^{b}=e^{4}\)
So
\(b=4\)
and
\(\ln e^{4} \quad[=b] \quad=4\)
Answer 2:
4
Explanation:
The logarithm of a power is the power × the logarithm of the number.
Example:
In \(a^{n} \Rightarrow n \ln e\)
So
In \(e^{4} \Rightarrow 4 \ln e\)
The logarithm of the base is always \(1\)
Proof:
In \(e=y \Rightarrow e^{y}=e \Rightarrow e^{y}=e^{1}\) (if bases are the same then powers are equal)
Example:
\(\log _{10}(10)=1\)
\(\log _{2}(2)=1\)
So \(\ln e=1\)
So we have:
\(4 \ln e \Rightarrow 4(1)=4\)