Discrete math, Big Theta notation?

Question

Would anyone know how to show that Log(base 10) x is big Theta (Log(base 2) x) if anything hints would be fine.

Answer 1

f(x) ∈ Θ(g(x)) ⇔ 0 < [x→∞]lim inf |f(x)/g(x)| ≤ [x→∞]lim sup |f(x)/g(x)| < ∞. We can prove both bounds at once by showing that [x→∞]lim |f(x)/g(x)| exists, and 0 < [x→∞]lim |f(x)/g(x)| < ∞, so let's do that:

[x→∞]lim |f(x)/g(x)|

Substituting the formulas:

[x→∞]lim |log x/log₂ x|

Use the change of base formula for logarithms:

[x→∞]lim |log x/(log x/log 2)|

Cancel:

[x→∞]lim |log 2|

And the limit of a constant is just a constant:

log 2

Since 0 < log 2 < ∞, we have log x∈Θ(log₂ x).

Answer 2

greater often than not, a functionality f(n) being O(g(n)) ability that, for all sufficiently great n, |f(n)| < c * |g(n)|. So, thus, you may tutor that (3n^3 - 4n^2 + 6n - 12) < cn^3 for all sufficiently great n. notice that f(n) would be beneficial for all sufficiently great n (that's a 0.33-degree polynomial functionality with a favorable n^3 coefficient). in case you divide the two facets with the help of n^3 ("sufficiently great" means that we in basic terms could evaluate beneficial values for n, letting us try this with no need to rigidity approximately probable having to tutor the less-than situation), you get 3 - (some stuff that gets closer to a persevering with with increasing n) < c All you certainly could do is detect a c for which that's real. 40 two does properly. i'm guessing that the guy who wrote the e book grow to be a Douglas Adams fan (truly, of the Hitchhiker's instruction manual to the Galaxy)