Derive the truncated Taylor series expansion and the respective error term

Question

For $f(x) = \log_2(2x)$

Derive the truncated Taylor series expansion and the respective error term when truncating the Taylor series for f(x + h) developed at x = 1 after the n-th term.

Any idea, hint or solution is welcomed.

Thank you

Answer 1

Hint:

$$\log_2(2x)=1+\log_2(e)\ln(x)$$

Now use well known expansions of $\ln(x)$ at $x=1$.