Question

What is the base of
In x
And log x ??

Answer 1

for ln(x), the base is e, the natural base (aprox equal to 2.718281828454590...) or
`ln(x)=log_e(x)`

for log(x), if no base is stated, assume base 10 or
`log(x)=log_(10)(x)`

Answer 2

On a standard scientific calculator, the log button denotes the "common logarithm", i.e.
`log_(10)`. This is consistent with the common usage in engineering and the natural sciences; for example, the pH scale used for measuring acidity, the Richter scale used for measuring earthquake intensity, and the decibel scale used for measuring sound intensity are all defined using a base-10 logarithm. Scientific calculators use the ln button to indicate the "natural logarithm", i.e.
`log_e`.

In contrast, mathematicians tend to use the symbol log to refer to
`log_e`. That's because (from the point of view of pure mathematics) there is nothing special about the number 10, and no real reason to define a logarithm to a a single, arbitrary privileged base. From a pure mathematics standpoint, the only logarithm that really matters is
`log_e`, so this is what the generic symbol log refers to.

For some reason many mathematicians tend to be oblivious to the fact that outside of our own tribe nearly everyone uses the symbols log and ln to refer to different things, and rather obstinate in insisting that log=
`log_e`, as if notation were not merely a convention but rather somehow a law of nature or a logical necessity. Notation is always conventional, and all conventions are local.