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?

In lower level mathematics, I normally take algebra to mean the study of relationships between functions and equations, and arithmetic to mean the study of the operations, such as addition, subtraction, multiplication, division.
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.
