We always heard about the logarithms log and ln. So it is important to know about their meaning and differences. Note that log represents the common logarithm and ln represents the natural logarithm. In this post, we will learn about them.
What is log?
The notation log is referred to the short form of the logarithm. log denotes the common logarithm and it is with base 10.
For example, $\log 2$ is the natural logarithm of $2$. That is, $\log 2=\log_{10} 2$.
As we know that the logarithm of $x$ with base $x$ is equal to $1$, that is, $\log_a a=1$, we must have that $\log_{10} 10=1$. Thus, the common logarithm of $10$ is equal to $1$.
What is ln?
The notation ln is used to denote the natural logarithm and it is with base $e$. Here, $e$ is the irrational number defined by the following convergent series:
$e=\sum_{n=0}^\infty \dfrac{1}{n!}$ $=1+\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+\cdots$
The value of $e$ is approximately equal to $2.71828$.
$e \approx 2.71828$
Difference between log and ln
| Log | Ln |
| log is the logarithm with base 10. | ln is the logarithm with base e. |
| It is also known as common logarithm. | It is also called natural logarithm. |
| Notation: $\log_{10} x$ | Notation: $\log_{e} x$ |
| Exponential Form: $10^x=y$ | Exponential Form: $e^x=y$ |
| It has more uses in Physics than ln. | ln is less used than log in Physics. |
Question Answer on log vs ln
Question 1: Find the value of e to the power lnx, that is, evaluate $e^{\ln x}$.
Answer:
Let us assume that $y=e^{\ln x}$. We need to find the value of $y$.
Takning $\ln$ both sides of $y=e^{\ln x}$, we have that
$\ln y = \ln (e^{\ln x})$
$\Rightarrow \ln y =\ln x \ln e$
$\Rightarrow \ln y =\ln x$ as we know that $\ln e=1$.
$\Rightarrow y = x$.
Thus, we have shown that the value of e to the power ln x is equal to x.