# In which branch of mathematics does “logarithm” belong? Arithmetic or algebra?

I'm currently working on an iOS & Android application for GCE O Level students. I have to classify everything neatly such that Maths never appears to be a messy subject to study and so

should I classify logarithm into arithmetic or algebra?

Here is the definition of arithmetic by Oxford dictionary:

the branch of mathematics dealing with the properties and manipulation of numbers.

and the definition of algebra:

the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations

According to this, there would be an inconsistency.

The usage of logarithm is considered arithmetic since it is manipulating number.

$$Log_{10} 100 = 2$$

And the laws of logarithms would be considered algebra.

$$Log_b (MN) = Log_b (M) + Log_b (N)$$

This really frustrates me because I will have to classify the usage of logarithm and the laws of logarithm into different branches of mathematics but they are both under the same group.

What should I do?

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If by algebra you mean pre-calculus algebra (vs abstract algebra) then I personally don't see the point of really separating arithmetic and algebra. The two have an enormous amount of overlap. –  EuYu Nov 26 '12 at 9:41
As much beauty as there is in mathematics, I don't know that I would say it, "is never a messy subject to study." –  Jeremy Nov 26 '12 at 9:43
@EuYu Yeah, I agree with you. But this is an application designed for secondary school students (grade 7 to grade 10) and they are just starting to learning about rational number, prime number, ratio, etc so it would be better to separate arithmetic and algebra. –  user38927 Nov 26 '12 at 9:47

I guess the main question then becomes, do we lump exponentiation and logarithms into arithmetic, or say that it belongs under algebra? My personal view is that, if we really have to draw the line somewhere (of course, the distinctions are terribly arbitrary), then logarithms and exponents should belong under algebra. The primary reason for this is that we are typically not interested in actually calculating exponents and logarithms; I think of elementary school arithmetic as the set of skills that students need to calculate quantities throughout their lives. One cannot, for example, calculate $\log{5}$ or $e^{2}$, in their heads, except through very rough approximation. Usually, it is the properties of logarithms that we are interested in studying, because they allow us to solve certain equations or rewrite certain functions, just as you have mentioned, and I believe this squarely makes the study of logarithms part of secondary school algebra.