What is the base of $\log x$?

Question

I've seen "$\log x$" being used in some papers (and by Wolfram|Alpha), and I was confused because so far I have only ever seen the $\log$ used with a base ( so e.g. $\log_y x$).

Am I correct that $\log x = \log_e x = \ln x$?

• If so, why is $\log x$ used over $\ln x$? Isn't the letter more expressive and less confusing?

• If not, what is the base of $\log x$?

Answer 1

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.

Answer 2

Usually in math context without any other specification the symbols $\log$ and $\ln$ are both used to denote the natural logarithm.

Refer also to Particular bases for logarithm

Answer 3

This depends heavily on context, but the most common definition is $\log x=\ln x$. That said, in introductory textbooks for students, $\log$ is often used to mean $\log10$, and in coding theory $\log$ often refers to $\log2$. In general though, unless specified it typically refers to $\log_e$.

As to why there's no standardised convention it's hard to say for certain, but most likely just historical reasons: authors picked convention to what was most convenient, and that 'stuck' within each field of mathematics. More details about this could be found here.