Proving that $(2^3)^{\log_2 n}=n^3$

Question

may someone help me understand why $$\large (2^3)^{\log_2 n}=n^3$$

I don't know if this helps but we know that: $$\large 2^{\log_2 n}=n$$

Answer 1

Remember $(a^b)^c=a^{bc}=(a^c)^b$, so $$ (2^3)^{\log_2 n}=(2^{\log_2 n})^3=n^3. $$

Answer 2

Hint: $(x^a)^b = x^{ab} = (x^b)^a$.

Answer 3

Quite simple: $$\bigl(2^3\bigr)^{\log_2 n}=\bigl(2^{\log_2n}\bigr)^3=n^3.$$

Answer 4

$$(2^3)^{\log_2n}=(2)^{3\log_2n}=(2)^{\log_2(n^3)}=n^3$$

Answer 5

I would like to mention a formula here that is pretty useful: $$a^{\log_bc}=c^{\log_ba}$$ Notice that I've simply switched $a$ and $c$. Proving this is straightforward. I'll leave it to you for practice. If you need a hint, you can ask in the comments.

Coming to the use of this formula in the question, Let's pick up from where you left off: $$2^{\log_2n}=n^{\log_22}=n^1=n \;\;\; .$$