The fundamental log relationship is,

10log x = x,

which simply restates the definition of the logarithm: "Log x is the number to which 10 must be raised in order to get the number x."

Starting with

10log x = x and 10log y = y,

we multiply to get,

[10log x][10log y] = xy.          Equ. (1)

For exponents one always has,

10a10b = 10a + b,

so the left side of Equ. (1) is

10log x + log y, which equals xy.

So the power to which 10 must be raised in order to get xy is the exponent of the left side here. But the underlined quantity is, by definition, log xy. Thus,

log x + log y = log xy.

QED