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Find the natural logarithm of 0?

Last updated date: 16th Jul 2024
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Hint: We first try to explain the use of logarithm and the general formulas of the logarithms. We prove it by contradiction where we assume ${{\log }_{b}}0=x$ and try to find the value of $x$ which is not possible due to the condition of $a,b>0$ for ${{\log }_{b}}a$.

Complete step by step answer:
The logarithm is used to convert a large or very small number into the understandable domain. For the theorem to work the usual conditions of logarithm will have to follow. If the base is not mentioned then the general solution for the base for logarithm is 10. But the base of $e$ is fixed for $\ln $. We also need to remember that for logarithm function there has to be a domain constraint.
For any ${{\log }_{b}}a$, the conditions are $a,b>0$ and $b\ne 1$.
We also prove it using contradiction. We assume ${{\log }_{b}}0=x$. We know the formula that if ${{\log }_{b}}a=p$ then ${{b}^{p}}=a$.
So, for ${{\log }_{b}}0=x$ we get ${{b}^{x}}=0$ which is possible only when $b=0$ which creates the contradiction.
Therefore, natural logarithm of 0 is not possible.

Note: Only logarithms for numbers between 0 and 10 were typically included in logarithm tables. To obtain the logarithm of some number outside of this range, the number was first written in scientific notation as the product of its significant digits and its exponential power.