Show directly, using the definitions of Big−O, Ω, and Θ that for all $a,\:b,\:>\:1,\:\log _a\left(n\right)\:∈\:Θ\:\log _b\left(n\right)$

Question

i tried this and i got

set n=2

set c= 2

set a + b = 1

$\log _a\left(n\right)\:\le \:c\:\cdot \:\log _b\left(n\right)$

$\log _a\left(n\right)\:\le \:\:\log _b\left(n^c\right)$

$\log _a\left(2\right)\:\le \:\:\log _b\left(4\right)$

$1\le4$

I'll be honest, I dont really know how to do this

Answer 1

You can simply use the change of base formula for logarithms: $$ \log_b(n)=\frac{\log_a(n)}{\log_a(b)}$$ and observe that $\log_a(b)$ is a positive constant if $a,b>1$.