simplify $3n - 3 * 2^{\log _{3}(n)}$

Question

How can I simplify this so I don't have a log in the exponent ? $3n - 3 * 2^{\log _{3}(n)}$

Answer 1

Call $3n - 3 * 2^{\log _{3}(n)} = y $

'Exponentiate' both sides by 3,

You end up with $ 3^y = \frac {3^{3n}}{3^{(3)(2)^{\log _{3}(n)}}}$

Can you go from there?

Going further,

$ 3^y = \frac {3^{3n}}{3^{(3)(3^{ \log_3 (2) \log_3 (n))}}} = \frac {3^{3n}}{3^{3n^{ \log_3 (2)}}}$

$ \ln(3^y) = 3n \ln(3) - 3n^{ \log_3 (2)} ln(3) = y \ln(3)$

$ y = 3n - 3n^{ \log_3 (2)}$

Answer 2

Maybe this will help: $2 = 3^{\log_{3}(2)}$, and so $2^{\log _{3}(n)} = 3^{\log_{3}(2)\log_{3}(n)} = n^{\log_{3}(2)}$, so in total:

$$ 3n - 3 \cdot 2^{\log _{3}(n)} = 3n - 3n^{\log_{3}(2)} $$