Q:

Substituting another variable into a closed form recurrence relation and writing equation in terms of that variable

Given the function: $$f(x) = 2^x - x - 1$$ and the variable $$t = 2^x - 1$$
  Rewrite $f(x)$ as a function of $t.$

I try to substitute $t$ into $f(x)$ and get:
$$f(x) = t - x$$
and then substitute $$\log_2(t + 1)$$ for $x$ and get:
$$f(\log_2(t + 1)) = t - \log_2(t+1)$$
At this point I am kind of lost and getting conflicting answers based on how I proceed for instance trying this:
$$f(\log_2(t)) = (t-1) - \log_2(t)$$
$$f(t) = t^{t-1} - t$$
Just doesn't seem right. 
Not sure what to try next. 

A:

The answer is hidden within the steps you showed here. 
To rewrite $f$ in terms of another variable $t$ will give you a new function $\tilde{f}$ like so
$$
f(x) = 2^x-x-1 = (2^x-1) - x = t - \log_2(t+1) =: \tilde{f}(t) \ . 
$$

