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.