# What is the base of $\log x$? [duplicate]

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 $\log x$?

• 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 $\log_{10}$, and in coding theory $\log$ often refers to $\log_2$. In general though, unless specified it typically refers to $\log_e$. – B. Mehta May 27 '18 at 21:42
• Nowadays, usually $\log x=\ln x$. In older books, it can be $\log_{10} x$, especially in books dealing with numerical computations, as decimal logs were used, with the help of tables of logarithms or slide rules, to do most numerical computations by hand. – Jean-Claude Arbaut May 27 '18 at 21:45
• For WolframAlpha you're right: $\log$ means $\ln$. In other context it depends. For example on physics the natural logarithm is usually denoted by $\ln$, and I think sometimes $\log$ denotes the logarithm on base $10$. For example here in Spain in the school $\log$ denotes presicely the logarithm in base $10$. However, I like to use this notation for the natural logarithm and I write $\log_x$ when the base is different from $e$. As a rule: if you see in the same reference $\log$ and $\log$ the first is referring to the decimal logarithm. Otherwise... who knows. – Dog_69 May 27 '18 at 21:46
• @ThomasFlinkow 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. – B. Mehta May 27 '18 at 21:46
• @B.Mehta Why not add that as an answer? – Noah Schweber May 27 '18 at 22:08

## 3 Answers

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.

• For this reason the “common logarithm” is often called the “decimal logarithm” $\ddot\smile$ – gen-ℤ ready to perish May 27 '18 at 22:16
• In addition, it might be nice to note $\log_e x$ has the nice property that its derivative is $\frac{1}{x}$, which is why it's typically chosen in pure mathematics over any others. – B. Mehta May 27 '18 at 22:16
• Thank you for pointing out base 10 is still currently more common in the physical sciences for obvious reasons. – fleablood May 27 '18 at 22:22

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.

• "The most common definition" requires context. If you look across the natural sciences and engineering, I think the most common definition is unequivocally that $\log$ denotes $\log{10}$. Why else would calculators maintain the distinction between $\log$ and $\ln$? – mweiss May 27 '18 at 22:28
• Even in mathematics, the standard convention (at least in the United States) at the level of introductory Calculus is that $\ln = \log_e$ and $\log = \log_{10}$. – mweiss May 27 '18 at 22:29
• @mweiss This is a fair response, my intention was to mean that in mathematics (since this is MSE after all) past the introductory level, the most common definition is $\ln$. In particular, since OP was asking specifically about mathematical papers and W|A, I felt this context was implied. – B. Mehta May 27 '18 at 22:30

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