TITLE: Clarify why all logarithms differ by a constant
QUESTION [6 upvotes]: One of the rules of logarithms that I've been taught is:
$\log_b(x)$ is equivalent to $\dfrac{\log_k(x)}{\log_k(b)}$.
Recently I've also seen another rule that says:
$\log_b(x)$ is equivalent to $\log_b(k)\log_k(x)$.
Are these equivalent (perhaps via some refactoring) ?

REPLY [6 votes]: HINT $\: $ The first equation is $\rm\ log_K\ $ of $\rm\ B^{log_B\ X}\ =\ X$ 
and the $\:$ second $\:$ equation is $\rm\  log_B\ $ of $\rm\ K^{log_K\ X}\ =\ X$