Question

Can anyone help out with question 5a and 5b?

Answer 1

To find the inverse, let's isolate x:

$y=k\log_2(x) \iff \dfrac{y}{k} = \log_2(x) \iff 2^{\frac{y}{k}} = 2^{\log_2(x)} \iff 2^{\frac{y}{k}} = x$

So, the inverse function is

$f^{-1}(x) = 2^{\frac{x}{k}}$

We know that the inverse gives 8 when evaluated at x=1, so

$f^{-1}(1) = 2^{\frac{1}{k}} =8$

Now, since $8=2^3$, we need

$\dfrac{1}{k}=3 \iff k=\dfrac{1}{3}$

So, the inverse function is

$f^{-1}(x) = 2^{3x}$

Which means that

$f^{-1}\left(\dfrac{2}{3}\right) = 2^{3\frac{2}{3}}=2^2=4$