Given $ax^n = 2^k$ is it possible to calculate $k$ if $a$, $x$ and $n$ are known

Question

For example, we know that:

$2.143483648 * 10^9 = 2^{31}$

If we were given:

$2.143483648 * 10^9 = 2^k$

How do we find $k$ ?

Answer 1

if we take the $\log_2$ of both sides, we get that $$\log_2(ax^n) =\log_2(2^k)$$ $$log_2(a)+\log_2(x^n)=k$$ $$k=\log_2(a)+n\log_2(x)$$ In your example, $a=2.143483648$, $x=10$, $n=9$, which means that $$k=\log_2(2.143483648)+9\log_2(10)$$ Which is approximately 31, as you expected.

Answer 2

hint

If $$x=2^k$$ then if we take logarithm of both sides,

$$\ln(x)=\ln(2^k)=k\ln(2)$$

thus

$$k=\frac{\ln(x)}{\ln(2)}=\log_{2}(x)$$