An early application of recursion can be found in the seventeenth century in John Napier’s...

An early application of recursion can be found in the seventeenth century in John Napier’s method of finding logarithms. The method was as follows:

start with two numbers n, m and their logarithms logn, logm if they areknown;while not done  for a geometric mean of two earlier numbers find a logarithm which is an arithmetic mean of two earlier logarithms, that is, logk =  (logn+logm)/2 for k = ;  proceed recursively for pairs (n, ) and (, m);

For example, the 10-based logarithms of 100 and 1,000 are numbers 2 and 3, the geometric mean of 100 and 1,000 is 316.23, and the arithmetic mean of their logarithms, 2 and 3, is 2.5. thus, the logarithm of 316.23 equals 2.5. The process can be continued: the geometric mean of 100 and 316.23 is 177.83, whose logarithm is equal to (2 + 2.5)/2 = 2.25.

a. Write a recursive function logarithm() that outputs logarithms until the difference between adjacent logarithms is smaller than a certain small number.

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b. Modify this function so that a new function logarithmOf() finds a logarithm of a specific number x between 100 and 1,000. Stop processing if you reach a number y such that y – x < ϵ for some ϵ.

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c. Add a function that calls logarithmOf() after determining between what powers of 10 a number x falls so that it does not have to be a number between 100 and 1,000.