The logarithmic function
The logarithmic
function is the inverse of the exponential (power) function. We can think of
this in terms of shorthand:
·
The
addition function is shorthand for the counting function
o 2 + 4 is the same as starting from 2 and
counting up four times
·
The
multiplication function is shorthand for the addition function
o 3 * 4 is the same as adding three to itself
four times
·
The
exponential function is shorthand for multiplication
o 34 is three multiplied by itself
four times
In fact, these can
be generalised into the Ackermann function, with shorthand for
exponentials, and shorthand for shorthand for exponentials etc. etc.
The logarithmic
function is a mathematical function that represents the inverse of the
exponential function. It is defined as the exponent to which a fixed base must
be raised to produce a given number.
For example, the
logarithm of 1000 to base 10 is 3, because 10^3 equals 1000. We can write this
as log_10(1000) = 3.
The logarithmic
function is commonly used in mathematics and computer science, and it has
several important properties:
1.
The
logarithmic function is an increasing function. This means that if a < b,
then log(a) < log(b).
2.
The
logarithmic function is the inverse of the exponential function. This means
that if y = log(x), then x = ey.
3.
The
logarithmic function satisfies the following property: log(xy) = log(x) +
log(y). This means that the logarithm of the product of two numbers is equal to
the sum of the logarithms of the numbers.
4.
The
logarithmic function has the following property: log(x/y) = log(x) - log(y).
This means that the logarithm of the ratio of two numbers is equal to the
difference of the logarithms of the numbers.
In computer
science, the logarithmic function is often used to measure the time complexity
of algorithms, because it grows very slowly as the input size increases. For
example, an algorithm with a time complexity of O(log n) will generally be much
faster than an algorithm with a time complexity of O(n), for large input sizes.
The Ackermann function is a mathematical function that is
defined recursively as follows:
Ackermann(m,n) = n+1 if m = 0 Ackermann(m,n) =
Ackermann(m-1,1) if m > 0 and n = 0 Ackermann(m,n) =
Ackermann(m-1,Ackermann(m,n-1)) if m > 0 and n > 0
The Ackermann function was originally introduced as an
example of a well-defined function that is not primitive recursive, meaning
that it cannot be computed by a set of basic arithmetic operations and the use
of a fixed number of nested loops.
The Ackermann function grows very quickly, even for small
input values. For example, the value of Ackermann(4,1) is over 13,000. As a
result, the Ackermann function is not used in practice, but it is still studied
in theoretical computer science as an example of a function with unusual
computational complexity.
One interesting property of the Ackermann function is that
it is not total, meaning that there are some input values for which it does not
return a result. For example, the function does not return a result for the
input values Ackermann(4,2) and Ackermann(5,0). This is because the function is
defined recursively, and the recursion goes on indefinitely for some input
values, leading to an infinite loop.
Each of the
shorthands on the previous sides has an inverse
·
Subtraction
is the inverse of addition
o If a + b = c, then c - a = b
·
Division
is the inverse of multiplication
o If a * b = c then c / a = b
·
The
logarithmic function is the inverse of the exponential function
o If ab = c then loga
c = b
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