# What is the difference between $ \log $ and $ \ln $ ?

Last updated date: 22nd Mar 2023

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**Hint**: In the above question, we are required to point out a difference between the given two logarithmic functions with different bases. So to solve this question requires theoretical knowledge regarding the logarithmic functions and its base. A logarithm function is the inverse of an exponential function (a function in which one term is raised to the power of another term is known as an exponential function). An exponential function is of the form $ a = {x^y} $ , so the logarithm function being the inverse of the exponential function is of the form $ y = {\log _x}a $ .

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__Complete step-by-step answer__In the given problem, we have to differentiate between the two mathematical functions $ \log $ and $ \ln $ provided to us in the problem itself.

So, the $ \log $ function is the logarithm function with base being equal to $ 10 $ . So, it can also be written as $ {\log _{10}} $ to be clear and understandable.

On the other hand, the $ \ln $ function is the logarithmic function with base being equal to e, where e is the Euler’s number or constant. So, it can be written as $ {\log _e} $ to be clear. This $ \ln $ function is also called a natural logarithm function.

The standard base of logarithm functions is 10, that is, if we are given a function without any base like $ \log x $ then we take the base as 10.

But, when we are specifically given the base of the logarithm function as e, we have to take the function as $ \ln x $ .

**Note**: So, both the functions given to us in the question are logarithmic functions just with different bases. We should know when to use which function as it can create a misunderstanding and confusion otherwise. We should keep in mind an important rule that the base of the logarithm functions involved should be the same in all the calculations in order to apply any property of logarithm. There are several laws of the logarithm that make the calculations easier and help us evaluate the logarithm functions.

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