Exponent or Logarithmic math

Amrish Macedo
3 min readMar 14, 2020

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Let us start with some basic understanding of really large to get you comfortable. I will start with time. We know time as seconds, minutes, hours, days, months etc. How many seconds are in a year? There are 60 seconds/minute * 60 minutes/hour * 24 hours/day = 86,400 seconds in a day. I will round this to about 100,000 which is a lot easier to remember than 86,400. If someone asks, now you can say there are around 100,000 seconds in a day. 1 followed by 5 zeros or 10⁵ seconds in a day. If we multiple 86,400 by 365 days in the year, there are about 31 million seconds in a year. Now we need a little more help for this.

In the table of exponent examples, you can see how everyday terms can be represented in exponent form. A million is shown as 10⁶ (it looks a little different because medium converts the carat symbol into the superscript representation). How can we use this to make big numbers easier to manage.

Logarithmic math

Is there an easier way to solve this problem. If we are not looking for too much precision, we can use logarithmic math, using a base 10. What does this mean? Multiplying anything by 10 is easy. Just place a zero to the right of the number. Seven multiplied by 10 [7*10] is seven with a zero to the right or seventy. What about 10 multiplied by 10. That is one hundred. If we do this using the exponents, we discover 10¹ * 10¹ is 10². It is adding the exponents. That makes things a lot easier, compared to multiplying. 10,000 [10⁴] * 1,000,000 [10⁶] is actually (adding the exponents) 4+6, which makes it 10¹⁰.

Back to seconds

How do we multiply 86,400 * 365 to get a reasonable response around 31 million? One could round up 86,400 to be 90,000 or 9*10⁴. We could also round up 365 to 400 or 4*10². Now we have to solve for (9*10⁴) * (4*10²). 9*4 is 36 and adding the exponents we get 10⁶. The number we get is 36*10⁶ or if we look up the table above that is 36 million. That is not 31 million, but quite close for a lot less work.

10⁶ is not twice the size of 10³

This is a really important point I have to make. We added the exponents to multiply two numbers. When we add two numbers that are the same we can safely assume the result is twice the size of the initial numbers. Just because we added the exponents, we cannot assume the same. 10³ is 1,000. Double of 1,000 is 2,000. Multiplying 1,000 by itself, by adding the exponents, gets us to one million or 1,000,000. This is clearly not double of 1,000. Please understand this.

I want to restate this point. If you look at the table above you will see a million is 10⁶. If you multiply a million by a million, you get a trillion or 10¹².

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