$\log_2$ approximation in $[1,2)$ this is realistically for a programming project, but is more math centric then CS centric. I am attempting to write a function that approximates a power function, but in order to complete I need to approximate log2(x). using the understanding that
$$x^y = x^{a+b+c+d+\dots} = x^a*x^b*x^c*x^d\dots || y = a+b+c+d+\dots$$
and that 
$\displaystyle x^y = 2^{y\log_2(x)}$ computers think in base $2$ easier then base $10$, or $e$
since this would only be used for values $y \in [1,2)$ the solution only needs to be accurate on those bounds. considering that $log_c(a*c^x) = log_c(a) + x$ by factorization (not necessarily prime) I can reduce the input to a value on those bounds.
I need the RHS of $\displaystyle log_2(x) = || x \in [1,2)$
defined in addition, subtraction, multiplication, and division. whole number exponents are also acceptable. 
EDIT: did some analysis, and updated question.
 A: I assume that you are working with floating-point numbers ala IEEE 754. In this case, I do not see how base $2$ is going to give you any significant advantage. Instead, I would use the identities
$$\ln(x) = 2\cdot\sum_{k=0}^\infty \left(\frac{x-1}{x+1}\right)^{2k+1} \cdot \frac{1}{2k+1}$$
and 
$$\exp(x) = \sum_{k=0}^\infty \frac{x^k}{k!},$$
both of which are very fast converging series. Then, $x^y = \exp(y\cdot\ln(x))$. See also the wiki page on computing the logarithm.
A: The CORDIC algorithm can be used to give good $\log$ and $\exp$ approximations. I wrote such functions using fixed point math for QuickDraw GX.  All that is required is shifting, adding, and subtracting.
A: $$log_2(x) = \sum_{k=0}^\infty \frac{-(-1)^K(x-1)^k}{\ln(2^k)},$$ convergent on $x \in [1,2)$ after about $10$ terms though may needs about 12 terms to be mostly convergent on those bounds, 
and for the cs application a table of the $\ln(2^k)$ values should suffice. though a table of $\frac{1}{\ln(2^k)}$ may be more beneficial depending on the actual values there in, and that multiplication by a floating point number is more efficient then division.
