Alg9.4.2

If a not equal to 1 and m,n are positive real numbers, then $\log_a(mn)=\log_a(m)+\log_a(n)\,$

Let $\log_a(m)=x,\log_a(n)=y\,$

Therefore

$m=a^x,n=a^y\,$

$mn=(a^x)\cdot(a^y)=a^{x+y}\,$

Taking log on both sides with base a, we get

$\log_a(mn)=x+y\,$

substituting the values of x and y

$\log_a(mn)=\log_a(m)+\log_a(n)\,$