You can see the two types of logarithm in most of cases, and they are:

1. Common Logarithm
2. Natural Logarithm

Common Logarithm(Base 10)

The common logarithm is also known as the base ten logarithms. It is written as \(p \log p\). So, when the logarithm is taken with respect to base 10, then we call it is the common logarithm. The base is sometimes not written in a common logarithm. In some logarithms, if the base is not written, we assume that it is a common logarithm with base 10.

\(\text { common logarithm is written as } \log x=\log _{10} x\)

Some examples are given below.

\(
\begin{aligned}
&\begin{array}{ll}
\log _{10} 10=1, & \text { since } 10^1=10 \\
\log _{10} 100=2, & \text { since } 10^2=100 \\
\log _{10} 10000=4, & \text { since } 10^4=10000 \\
\log _{10} 0.01=-2, & \text { since } 10^{-2}=0.01 \\
\log _{10} 0.001=-3, & \text { since } 10^{-3}=0.001 \\
\text { and } \log _{10} 1=0 & \text { since } 10^0=1
\end{array}
\end{aligned}
\)

Natural Logarithm (Base e)

The natural \(\log (\ln )\) is the inverse operation of \(e\), the natural exponent. \(\text { natural logarithm is written as } \ln x=\log _e x\)

\(
\begin{aligned}
& e^y=x \\
& \ln (x)=y 
\end{aligned}
\)

Some examples are given below:

\(2.71828^1=2.71828 \Rightarrow e^1=2.71828 \Rightarrow 2.71828=1\), or, \(e=1\) or simply \(\ln e=1\)
\(2.71828^2=7.39 \Rightarrow e^2=7.39 \Rightarrow 7.39=2\), or, \(7.39=1\) or simply \(\ln 7.39=2\)
\(2.71828^3=20.08 \Rightarrow e^3=20.08 \Rightarrow 3\), or, \(20.08=3\) or simply \(\ln 20.08=3\)
and, so on.