Why is b>1 and x,y positive in logarithm definition?

[UchihaClan13]

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]

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]

X and Y must be positive because if log_A(X) = B, then A^B=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]

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.

 

[member 587159]

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.

 

[micromass]

This is not a definition.

It can be.

 

[Googlu02]

Thats just definition of logarithms