given two function:
$a\:\le\:b\cdot log\left(a\right)\:$
$a\:\le\:b\cdot \:log\left(b\right)\:$
i want to show that $b\cdot \:log\left(a\right)\:\le \:b\cdot log\left(b\right)\:$
I made a graph of these 2 functions, you can clearly see that this is true, at infinity, this equation (2) is always above this equation (1)
I was thinking of showing the limit at infinity: $\lim _{a\to \infty }\left(\left(b\cdot \:log\left(b\right)\:\right)/\left(\:b\cdot \:log\left(a\right)\:\right)\right) = 0$
I'm not sure if this is a correct way to do it, am I right?