I was solving an integral and I encountered in some question
$$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$
I know its integration is $\log(x)$. But my answer comes correct when I use $\ln(x)$ instead. What is this confusion? How do I know which one to use? Thanks
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$\begingroup$$\log(x)$ has different meanings depending on context. It can often mean:
- any logarithm if the base is not important (e.g. in general proofs about logarithms)
- the natural logarithm $\log_e(x) = \ln(x)$ (usual convention in mathematics)
- the decadic logarithm $\log_{10}(x)$ (usual convention in chemistry, biology and other sciences)
- the binary logarithm $\log_{2}(x)$ (most often used in computer science)
In your case, $\log(x)$ just means the same as $\ln(x)$
$\endgroup$ 7 $\begingroup$The notation $\log$ is used for logarithms in general. For specific logarithm bases you normally use $\ln$ (for natural logarithm), $\lg$ (for base 10). Sometimes $\operatorname{lb}$ is used for base two.
So the question is what base is meant when writing $\log$. Normally one would indicate the base by subscribing the base. For example $\lg x = \log_{10} x$ or $\ln x = \log_ex$.
The problem then is what you make of an expression of $\log x$? That's ambiguous since the base is not specified. However in some cases it would be assumed that the default base is $e$ (but in some cases it maybe different).
Bottom line is if you don't want confusion you should normally use $\ln$ instead, or maybe $\log_e$ and never use $\log$ without subscript.
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