Iterative calculation of $\log x$ Suppose one is given an initial approximation of $\log x$, $y_0$, so that:
$$y_0 = \log x + \epsilon \approx \log x$$
Here, all that is known about $x$ is that $x>1$. Is there a general method of improving that estimation using only addition & multiplication, i.e. without exponentiation or logarithms?
$$y_1 = f(y_0, x)=\ ?$$

In any case, you also need to know $x_0=e^{y_0}$.
Then the correction is given by $y-y_0=\ln\left(\dfrac x{x_0}\right)=\ln(1+\delta)$, where $\delta$ is small. You can evaluate it by Taylor which will converge linearly,
$$\ln(1+\delta)\approx\delta-\frac{\delta^2}2+\frac{\delta^3}3\cdots$$
You can also limit yourself to the first term
$$\ln(1+\delta)\approx\delta,$$
but then you will have to correct with $x_1=x_0e^{-\delta}\approx x_0\left(1-\delta+\frac{\delta^2}2\cdots\right)$, which converges a little faster.
I don't think there is a shortcut.

A similar approach is given by the CORDIC method.
Precompute a number of decreasing constants $l_k=\log(a_k)$, for instance $a_k=1+2^{-k}$.
You will apply the following steps for all $k$ in turn:
$$\text{while } a_kx_n<x\to x_{n+1}=a_{n+1}x_n, y_{n+1}=y_n+l_k.$$
(If $a_{k+1}>\sqrt{a_k}$, an $\text{if}$ suffices.)
