# Logarithm Sign Convention

## Main Question or Discussion Point

A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms
we define
$$\log_b(xy) = \log_b(x) + \log_b(y)$$
Here
why is the condition imposed that b>1 and b is not equal to zero and that x and y are positive numbers?
Is it something to do with the function being continuous and monotonically increasing or decreasing in certain intervals(1,infinity) and (0,1) respectively?

UchihaClan13

mfb
Mentor
I fixed the formula, the image didn't get displayed.

You need the three terms to be defined to have an equation. Unless you introduce complex numbers, the logarithm is not defined for negative numbers, and a zero or negative base doesn't make sense, and b=1 doesn't work either. A base between 0 and 1 would be possible, but odd.

Drakkith
Staff Emeritus
X and Y must be positive because if logA(X) = B, then AB=X. Since you cannot raise A to any power and get a negative number (except possibly with complex numbers, not sure) X must be positive. The same applies for Y.

fresh_42
Mentor
For $b < 1$ one gets $\log_b x = - \log_{\frac{1}{b}} x$ and end up with a basis above $1$.
Thus there is simply no need to consider basis below $1$. And of course $b=1$ cannot be defined at all.

Math_QED
Homework Helper
2019 Award
A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms
we define
$$\log_b(xy) = \log_b(x) + \log_b(y)$$

UchihaClan13
This is not a definition.

This is not a definition.
It can be.

Thats just definition of logarithms