Getting the limit for a ratio of logarithms functions with different base

Question

I know similar questions have been asked in the past, but I need some clarification to know if what I'm doing is correct:

let $f(x) = log_2x$, and $g(x)= log_3x$

Now I need to take the limit as $x$ approaches $\infty$ from the ratio:

$f(x)/g(x)$

I have applied the base change formula for $g(x)$ which yielded $log_2x/log_23$

now, when I divide $log_2x/log_2x/log_23$ I get:

$log_23$

so, how would I go about the taking the limit of $log_23$ since I could not find a rule regarding logarithms so far. and I'm doing this exercises as a part of algorithms class I'm taking.

Answer 1

Taking the limit here doesn't involve any specific rules regarding logarithms. This is because since $\log_{2}3$ is a constant, i.e., doesn't involve $x$ at all, the limit of it as $x \to \infty$ would be the same value, i.e., $\log_{2}3$.