This is a very usual confusion and the answer does not have a consensual approach. In some school books these two logarithms are shown through different designations. As far as we are concerned, they are different designations applied to different things!
The logarithm of `x`, whose base is number `e`, is called natural algorithm and it can be figured out according to two different ways:
The "natural" base, which sometimes has the designation of Euler Number, has nearly the following value: ` e = 2,71828...`
The Napierian logarithm has this designation thanks to the Scottish mathematician John Napier, who has used the logarithm with the base `1/e`. Therefore, it is obvious that `log_e x != log_(1/e) x`, and so the natural logarithm and the Napierian logarithm are different concepts.
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