approximation

[mcgrim]

I am writing a function that approximates the natural logarithm function by using the following formula
under iteration:
a sub 0= (1+x)/2 b sub 0= sqrt(x)
a sub (i+1)= (a sub i + b sub i)/2
b sub (i+1) = sqrt(a sub (i+1) + b sub i)

and the approximation to ln(x) is equal to ((x-1)/a sub i)

this is the code I came up with but I am not sure I am getting the right results,
since when I increase n, the approximation does not get closer to ln(x) (shouldn't it? I'm not sure, if you know,
please share), and the graphs I am getting are empty.

def applog(n,x):
    a0=(1+x)/2
    b0=sqrt(x)
    for i in range(n):
        a0=(a0+b0)/2
        b0=sqrt((a0+b0/2)*b0)
    return (x-1)/a0
 

print("the log approximation is: " , ((applog(1,4)))) 
print("log value-------------------:" , (math.log(4)) )
                       #of n and get strange result
error=(abs( applog(4,4)- (math.log(4))   ))                    
print("the error is : ", error)   
    
        

plt.plot((applog(1,4)) )
plt.plot((math.log(4)))
plt.show()          #should plot both funtctions in the same graph
plt.plot(error )
plt.show()         # should plot the difference of both functions