Logarithms play an essential role in numerical computations. First, what are they?
If you multiply a number once by itself, you obtain a power of that number.
Typically, 2 * 2 = 22.
From this it follows that if you multiply a number, x, by itself, n times, you obtain the power xn.
The number x is called the root and the number n is called the exponent. Together, xn is called a power. It’s also very important to name results of operations, hence, let’s call the result of raising x to the n -th power, y.
This implies that xn = y. The number y can now be seen as the power xn, by virtue of equality. A practical example is 23, which is 8. The investigator is always busy searching for unknowns. Suppose we are given an incomplete power and are demanded to analyze it and determine the unkown.
Suppose instead of stating that 23 is 8, we are asked the question: what is 23?
The response is easy, we say, 23 = 8.
What if we are asked, what is the exponent x, such that 2 raised to x would yield 8?
That is if 2x = 8, what is the value of x ?
The algebraist would say, x is 3. The number 8 can be expressed as 23. Hence, the relation above can be written as 2x = 23. The number 8 has been replaced by its equivalent 23. Comparing both sides, we gain insight that the left hand side can be resolved into 8 by substituting 3 for x. Therefore, x = 3, and that’s the result.
The word logarithm can be taken to mean the synonym for exponent. We’ll see how. Suppose we are given the power 23 and are asked to name its parts. We’ll say: 2 is the root, and 3 is the exponent.
The logarithm of the power 23 is 3. Don’t ask me how.
However, the logarithm of a power or a number is the exponent to which the root or the number must be raised in order to get the resulting value represented by the power. Interestingly, that number is also called the exponent.
It follows that the logarithm of 23 is 3.
To generalize this, let log stand for logarithm and it will follow from this that log(xn) = n.
Also, xn = y, hence, log(xn) = log(y) = n.
Summarily, in the entire universe of numbers, every integer can be represented as a logarithm. The logarithm of any integer can be expressed as a decimal fraction. Since, our natural base for arithmetic is 10, every integer can be expressed as a logarithm in the base of 10.
Let’s investigate further. The question will be, what number can we raise 10 to, in order to obtain 10 itself?
The answer is 1.
The number 10 raised to the power 1 is 10.
Hence, log(101) = 1.
Logarithms are not defined for negative numbers, and this throws some light on the development of logarithms of numbers in base 10.
Precisely, we can ask the question, how does the logarithm behave for numbers greater than 0, but less than 10, and how does it equally behave for numbers greater than 10?
We know that log(10) = 1.
Hence, the logarithms of numbers less than 10 should be less than 1, and the logarithms of numbers greater than 10 should be greater than 1.
Similarly, numbers less than 10 have a single digit and numbers greater than or equal to 10 have multiple digits.
Back to the assertion that logarithms can be expressed using decimal fractions. Decimal fractions often have 2 parts: an integral part which appears before the “.”, and the fractional part which appears after.
The integral part of a decimal fraction in the case of logarithms is called its characteristic. It is this characteristic which will show us one beautiful application of logarithms.
log(1) = 0
log(2) = 0.30102999566
The characteristic of log(2) is 0
Be it as it may, we know and we can see that all the numbers less than 10 will have a characteristic of 0. And we can conclude that every single digit number has a characteristic of 0. We can further conclude that a characteristic of 0 implies that a number in question is single-digit.
How about numbers greater than 10?
log (12) = 1.07918124605
log (99) = 1.9956351946
log (123) = 2.08990511144
log (321) = 2.5065050324
So far, single digit numbers have a characteristic of 0, double digit numbers have a characteristic of 1, and triple digit numbers have a characteristic of 2.
Is there a pattern in this development?
Let’s test for numbers with quadruple, quintuple, sextuple and septuple digits.
log(1234) = 3.0913151597, 4 digits with a characteristic of 3.
log(12345) = 4.09149109427, 5 digits with a characteristic of 4.
log(123456) = 5.09151220163, 6 digits with a characteristic of 5.
log(1234567) = 6.09151466409, 7 digits with a characteristic of 6.
The pattern is, an n-th digit number has a characteristic of n - 1. And we can also state that: a characteristic of n asserts that the integer in question has n + 1 digits.
Now, what is the application of this finding?
Well, it’s easy. Logarithms can be used to find the number of digits of a given integer. It may be wise to take the absolute value of integers in performing such operations since logarithms are undefined for negative numbers. but negative integers also have a number of digits. However, since negative integers have digits, we can ignore the sign and compute the length of the digits smoothly. The absolute value function comes in handy in this case.
This finding becomes very essential in computer programming. One can, therefore, write a program that computes the length of digits of an integer by computing the logarithm, taking the characteristic of the result and adding 1 to it.
The sum returned will be the length of digits in the given number. I’d say that one can simply take the result of flooring the logarithm of the given number.
Below is a python program that computes the length of digits of its integral arguments.
import math
"""
numDigits receives an integer as an argument and returns its number of digits.
"""
def numDigits(n):
num = abs(n)
num_log = math.log(num, 10)
num = int(num_log) + 1
return num
# now let's run a few examples
for i in range (10):
n = (i + 1) * (10 ** i)
print "%d has %d digits" % (n, numDigits(n))
1 has 1 digits
20 has 2 digits
300 has 3 digits
4000 has 4 digits
50000 has 5 digits
600000 has 6 digits
7000000 has 7 digits
80000000 has 8 digits
900000000 has 9 digits
10000000000 has 11 digits
And that is an application of logarithms.
Now that we are, we can show the other method for computing the another technique for computing the length of digits of a given number. This method relies on successive integer divisions and subsequent manipulation of remainders.
import math
"""
A variant of the numDigit function.
This one uses a chain of recursive calls.
"""
def numDigits2(n):
num = abs(n)
if num < 10:
return 1
num = num // 10
return 1 + numDigits2(num)
# now let's run a few examples
for i in range (10):
n = (i + 1) * (10 ** i)
print "%d has %d digits" % (n, numDigits2(n))
1 has 1 digits
20 has 2 digits
300 has 3 digits
4000 has 4 digits
50000 has 5 digitsy
600000 has 6 digits
7000000 has 7 digits
80000000 has 8 digits
900000000 has 9 digits
10000000000 has 11 digits
And that’s that. You can compare, at leisure, the efficiency of each of the methods.
Thanks for reading.