What exactly is a Logarithm?

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Anybody related to the field of Physics, Math, Computer Science, Engineering, etc. must be familiar with logarithms. But to the layman out there, what exactly is a logarithm?

Let us decipher.

If I gave you two large numbers and asked you to add them, you would do it without much problem, but what if I told you to multiply them? You would do it, of course, but it won’t be without any head-scratching. Everybody knows that addition is comparatively easier than multiplication!

Back in the 16^th century, there were no electronic calculators and large and complex calculations were extremely hard to do, if not impossible. This was restricting the growth of science, in general.

John Napier, a Scottish landowner who was also an astronomer, mathematician, and physicist thought of finding a method that would convert the multiplication problems into addition problems.

He must have been a very dedicated man, as he spent around 20 years of his life inventing something that he would call as logarithms. In 1614, he published Mirifici Logarithmorum Canonis Descriptio which introduced to the world the concept of log tables.

We won’t go into details as to how he devised the log tables, instead, we shall concern ourselves with their real meaning that any person with a non-mathematical background walking on the road would also understand.

The question that we intend to answer today is: what does log_ax = b really mean?

The above expression may be interpreted as: 

How many times should I multiply a by itself to obtain x? 

कितने a आपस में multiply होकर x बनाएंगे?

The answer is b! b tells कितने a आपस में multiply होकर x बनाएंगे|

Let us study by an example, consider log₂8 = 3.

This expression asks: how many times should I multiply 2 by itself to get 8?

The answer is 3 [2x2x2 = 8]. The attentive ones must have already caught that logs are nothing but powers.

The expression can also be phrased as:

What power of a gives x?

[e.g. 2³ gives 8]

So,

log_ax = b is exactly same as a^b = x 

It is like asking the same question but in a different way. 

The above equivalence relation also tells us that logarithms are shortcuts for exponents in the same way as :

1.     Multiplication is a shortcut for addition (3x5 = 5+5+5)

2.     Exponents are a shortcut for multiplication (4³ = 4x4x4)   

But does logarithm really convert a multiplication problem into an addition problem? 

Let us observe by a small example. Let us try to obtain the result of 2x3 using logarithms. Any one of you would tell me in a jiffy that the answer is 6. But do logarithms really produce the same result?

Let’s first find out the values of log₁₀2 and log₁₀3.

(For the sake of simplicity, we take up to 3 places of decimals.)

        log₁₀2 = 0.301

        log₁₀3 = 0.477

Adding these two,

        log₁₀2 + log₁₀3 = 0.778.

Now, look for 0.778 in the anti-log table and you would find it adjacent to 5.999!

The astute observer might be wondering what 10 signifies in the log expression.

In the expression log_ax, a is known as the base.

For manual calculations base 10 is used because calculations in the factors of 10 are more intuitive to humans. It is different from the base that John Napier used while creating the log tables.