I came across logarithms in my 9th grade. For a long time I never really understood its use in real life. Logarithm of a number is simply the power to which 10 (or any other base) must be raised to equal the number in question.
log10(100) is 2 because 102 = 100 log10(1000) is 3 because 103 = 1000 log10(10000) is 4 because 104 = 10000
In the book The Joy of X – Steven Strogatz writes
Notice something magical here: as the numbers inside the logarithms grew multiplicatively, increasing tenfold each time from 100 to 1,000 to 10,000, their logarithms grew additively, increasing from 2 to 3 to 4. Our brains perform a similar trick when we listen to music. The frequencies of the notes in a scale – do, re, mi, fa, sol, la, ti, do – sound to use like they’re rising in equal steps. But objectively their vibrational frequencies are rising by equal multiplies. We perceive pitch logarithmically.
Here is an excellent article which explains the basics of logarithms. Let us look at some of its uses in real life.
1. Safety Index
Consider the following statistic in the US
1 in 5,300 dies each year due to car crash. 1 in 800 dies each year due to diseases caused by smoking. 1 in 2,000,000 is killed by lightning.
Let us take logarithm for all these number
log10(5300) = 3.7 [car crash] log10(800) = 2.9 [smoking] log10(2000000) = 6.3 [lightning]
If one in X person die as a result of doing some given activity each year, the safety index for that activity is simply the logarithm of X. Higher the safety index, the safer the activity in question. Logarithms helps to shrink the numbers of very high magnitude to a smaller one which our brains can deal with easily.
2. pH value
pH is an abbreviation for power of hydrogen. The pH scale measures how acidic or basic a substance is. It ranges from 0 to 14. A pH of 7 is neutral (water). A pH less than 7 is acidic, and a pH greater than 7 is basic.
The acidity depends on the hydrogen ion concentration in the liquid (in moles per liter) written as [H+]. The greater the hydrogen ion concentration, the more acidic the solution. It is defined as
pH = -log10[H+]
Pure water contains hydrogen ion concentration of 1 * 10-7 moles. To calculate the pH
pH = -log(1 * 10-7) = -log(1) - log(10-7) = 0 - (-7) = 7
Which is easier for human brain to deal with 1 * 10-7 moles or pH value of 7? Clearly 7 is easier for our brain to handle. Thus logarithms helps us to deal with numbers of very small magnitudes. Given below is the pH values for different substances.
3. Binary Search
In computer science binary search is a good example of an O(log n) algorithm. In the book The Algorithm Design Manual – Steve Skiena writes
To locate a particular person p in a telephone book containing n names, you start by comparing p against the middle, or (n/2)nd name, say Monroe, Marilyn. Regardless of whether p belongs before this middle name (Dean, James) or after it (Presley, Elvis), after only one comparison you can discard one half of all the names in the book. The number of steps the algorithm takes equals the number of times we can halve n until only one name is left. By definition, this is exactly log2(n). Thus, twenty comparisons suffice to find any name in the million-name Manhattan phone book! Binary search is one of the most powerful ideas in algorithm design. This power apparent if we imagine being forced to live in a world with only unsorted telephone books.
Take a look at the table below. Binary search on 10 trillion items can be finished in around 44 steps.
Exponents and Logarithms are inverses of each other. Any time you want to better understand numbers of very large and small magnitudes make use of logarithms. In life exponents (compound interest) is your friend when it comes to investing money. Logarithms is your friend when it comes to spending it.