Question

What is the natural log of 2?

Answer 1

Hint: This type of question depends on the concept of natural logarithm. We know that the logarithm function is the inverse of exponential function. Also we know that the natural logarithm is the logarithm with the base equal to a mathematical constant \[e\]. The natural log of any number say \[c\] is represented as \[{{\log }_{e}}c\].

Complete step by step solution:
Now we have to find the value of the natural log of 2.
As we know that the natural logarithm is the logarithm with the base equal to a mathematical constant \[e\]. We can write the natural log of 2 as \[{{\log }_{e}}2\].
\[\Rightarrow {{\log }_{e}}2\] is the power of \[e\] needed to be equal to 2.
We suppose that the power of \[e\] needed to be equal to 2 is \[x\].
\[\Rightarrow {{e}^{x}}=2\]
Substituting value of \[e\] we get,
\[\Rightarrow {{2.718}^{x}}=2\]
To obtain the value of \[x\] applying logarithm to the base 10 we can write,
\[\Rightarrow {{\log }_{10}}\left( {{2.718}^{x}} \right)={{\log }_{10}}\left( 2 \right)\]
By applying the rules of logarithm, we get,
\[\begin{align}
  & \Rightarrow x{{\log }_{10}}2.718=0.3010 \\
 & \Rightarrow x\left( 0.4342 \right)=0.3010 \\
 & \Rightarrow x=\dfrac{0.3010}{0.4342} \\
 & \Rightarrow x=0.6932 \\
\end{align}\]
Hence, we can say that \[e\] raised to 0.6932 is equal to 2.

Thus the natural log of 2 is approximately equal to 0.6932.

Note: In this question students have to take note about the different logarithmic values such as the value of \[e\] is approximately equal to 2.718 and the value of logarithm of 2 to the base 10 is 0.3010. Also the value of logarithm of \[e\] to the base 10 that is the value of logarithm of 2.718 to the base 10 is approximately equal to 0.4342. Also students have to note that the value of natural log of 2 is the power of \[e\] needed to be equal to 2.