In both exponential functions and logarithms, any number can be the base. However, there are two bases that are used so frequently that mathematicians have special names for their logarithms, and even scientific and graphing calculators include keys specifically for them! These are the common logarithm and natural logarithm.

The difference between $\text{log}$ and $\text{ln}$ is that $\text{log}$ is defined for base $10$ and $\text{ln}$ is denoted for base $e$. For example, $\text{log}$ of base $2$ is represented as $\text{log}_2$ and $\text{log}$ of base $e$, i.e. $\text{log}_e = \text{ln}$.

Let’s understand the difference between common logarithm and natural logarithm with examples.

## What is a Logarithm?

A logarithm is nothing but another way of expressing exponents. These are used to solve problems that cannot be solved using the concept of exponents only.

Logarithm and exponent are inverse forms of each other. For an expression $b^{x} = a$, the equivalent logarithmic expression will be $\log _b a = x$.

Here, “$\log$” stands for logarithm and is read as “**Logarithm of a to the base b is equal to x**“.

## What is a Common Logarithm?

The logarithm of a number to the base $10$ is known as a common logarithm. This system was first introduced by Henry Briggs. This type of logarithm is used for numerical calculations. The base $10$ in common logarithm is usually omitted while writing the expression.

For example, $x = \log_{10} 2$ is written as $x = \log 2$, which is same as $10^{x} = 2$.

## What is Natural Logarithm?

The logarithm of a number to the base $e$ (Euler’s constant) is known as a natural logarithm. This system was first introduced by John Napier and hence is also known as the Napierian logarithm. This type of logarithm is used for numerical calculations. If the base $e$ in the natural logarithm is omitted while writing the expression, it is written as $\ln x$.

For example, $x = \log_e 2$ is written as $x = \ln 2$, which is the same as $e^{x} = 2$.

## Examples

**Ex 1:** Which of the following statements is incorrect?

(a) $\text{log} \left(1 + 2 + 3 \right) = \text{log } 1 + \text{log } 2 + \text{log } 3$

(b) $\text{log } \left(2 + 3 \right) = \text{log }\left(2 \times 3 \right)$

(c) $\text{log}_{10} 10 = 1$

(d) $\text{log}_{10} 1 = 0$

The correct statement among the given statements is (b) $\text{log } \left(2 + 3 \right) = \text{log }\left(2 \times 3 \right)$.

**Ex 2:** Find the value of $\text{log } 9$, when $\text{log } 27$ equals $1.431$.

$\text{log } 27 = 1.431 => \text{log } 3^3 = 1.431 => 3 \times \text{log } 3 = 1.431 => \text{log } 3 = \frac{1.431}{3} => \text{log } 3 = 0.477$

Now, $\text{log } 9 = \text{log } 3^2 = 2 \times \text{log } 3 = 2 \times 0.477 = 0.954$

**Ex 3:** What is the value of $\text{log}_2 10$, when $\text{log}_{10}2 = 0.3010$?

$\text{log}_2 10 = \frac{1}{\text{log}_10 2} = \frac{1}{0.3010} = $

**Ex 4:** What is the Value of $\text{log}_{10} 80$, When $\text{log}_{10} 2 = 0.3010$?

$\text{log}_{10} 80 = \text{log}_{10} (8 \times 10) = \text{log}_{10} 8 + \text{log}_{10} 10$

$=\text{log}_{10} 2^3 + 1 = 3 times \text{log}_{10} 2 + 1 = 3 times 0.3010 + 1 = 0.903 + 1 = 1.903$

## Difference Between Common Logarithm and Natural Logarithm

These are the differences between common logarithms and natural logarithms.

Log (Log Base 10) | Ln (Log Base e) |

Log refers to a logarithm to the base 10 | Ln refers to a logarithm to the base e |

It is also called a common logarithm | It is also called a natural logarithm |

The common log is represented as $\text{log}_{10}x$ | The natural log is represented as $\text{log}_ex$ |

The exponent form of the common logarithm is $10^x =y$ | The exponent form of the natural logarithm is $e^x =y$ |

The interrogative statement for the common logarithm is “At which number should we raise 10 to get y?” | The interrogative statement for the natural logarithm is “At which number should we raise Euler’s constant number to get y?” |

## Applications of Logarithms

Some of the real-life applications of logarithms are:

- They are used for measuring the magnitude of earthquakes. $\log_{10} E = 11.8 + 1.5 M$ (where $\log$ refers to the logarithm to the base 1$0$, $E$ is the energy released in $ergs$ and $M$ the Richter magnitude).
- Logarithms are used for measuring the noise levels in $dBs$ (decibels). $L_p = 10 \times log_{10} \left(\frac{p}{p_{0}} \right) dB$, where $p$ = Measured Power, $p_0$ = Reference Power
- They are used to measure the $pH$ level of chemicals. $pH = log_{10}[H^{+}]$, where $H^{+}$ is concentration of molar hydrogen ion concentration
- Logarithms are used in radioactivity, mainly to detect the half-life of a radioactive element. $t_{\frac{1}{2}} = \frac {\ln \left(2 \right)}{c}$, $t_{\frac {1}{2}}$ is half life of decaying quantity and $c$ is decaying constant
- Logarithms are used to measure exponential growth or exponential decay. The best examples of these can be:
- The growth of money at a fixed rate of interest, say, for example, you have ₹$10,000$ in your bank account at an interest of $2\%$. With the help of logarithms, you’ll be able to know when your money’s going to reach ₹$12,000$.
- The growth of bacteria on a Petri dish. If you have a petri dish having bacteria taking up around $0.1\%$ space of the dish and you also know the fact that they divide every $30$ minutes, you’ll be able to calculate the time by which the bacteria will fill up that entire dish through the use of logarithms.
- Logarithms are used to measure radioactive decay in radiocarbon dating. Radiocarbon dating is the method of determining the age of an organic object by implementing the properties of radiocarbon (Carbon-$14$, $^{14}C$)
- Logarithms are used in specific calculations where multiplications are turned into additions.
- Logarithms are also implemented to calculate the exponential growth of the population.
- Logarithmic calculations also arise in calculus. Such calculations are used for several calculations in the real world.
- Logarithms are used in combinatorics problems. Combinatorics is a specific branch of mathematics that’s concerned with the study of finite discrete structures.

## Conclusion

Logarithms, both common and natural have immense applications in real-life problems. Logarithms help us to simplify the complex calculations that appear while solving problems.

## Practice Problems

- Solve the equations
- $2^{x} + 2 = 10$
- $e^{2x} = 9$ (Take $ \ln 3 = 1.0986$)
- $e^{3x} = 8$ (Take $ \ln 2 = 0.6931$)

- Express $3^{x} \left(2^{2x} \right) = \frac{144}{25} \left( 5^{x} \right)$ in the form $a^{x} = b$. Hence, find $x$.
- Solve without using a calculator: (Use $ \log 2 = 0.3010$, $ \log 3 = 0.4771$, $ \ln 3 = 1.0986$, $ \ln 5 = 1.6094$)
- $ \log3^{3} $
- $ \log 1$
- $ \log16^{2} $
- $ \ln e^{3} $
- $ \log 3 $
- $ \log 3^{2} $
- $ \ln \sqrt {5} $
- $ \ln 729$

- Solve, round to four decimal places.
- $ \log x = \log 2x^{2} – 2 $
- $ \ln x + \ln \left( x + 1 \right) = 5$

- How many digits are there in $264$ when $\text{log} 2 = 0.30103$?