Heya!! Mayank here again!! It is gonna be an informative and interesting post, so let’s not waste our time and start… Try to answer my question:

CALCULATE THE NUMBER OF DIGITS IN 2⁴³ ??

Think and answer…… Mhmmmm…… Let’s give some more time……. Done?? If not don’t worry because you will learn it now!! The first reaction which comes to you is: “2⁴³!!!! OMG!!! It’s such a large large large number…  Has mayank gone insane??” 2⁴³ is indeed a very large number but of course I won’t give you such an insane task!!! There’s a concept behind it… The concept of “LOGARITHMS” !! Maybe most of you don’t know what is this logarithm so before explaining the question we will learn about logarithm…

LOGARITHM: THE MOTHER OF EXPONENTS

Let there be numbers a, n and m where.. aⁿ = m Then n is defined as n = loga m or logarithm of m to the base a where m>0, a>0,a≠0 For example: We know that 25 = 32!! Therefore we can say that 5= log2 32 or 5 is equal to logarithm of 32 to the base 2!! Log is just the short form of logarithm… There are many many properties on this small concept!!

SOME IMPORTANT PROPERTIES OF LOGARITHMS:

1. The logarithm of any number n to the base 10 is called a common logarithm..  Ex: log10 n  In common logarithms we usually don’t write the base 10 and therefore if you see in any log with no base written just understand it has a base 10.. Therefore in the previous example it can be written as: log n 2.  Logarithm of any number n to the base e is called a natural logarithm.. For example: loge n Here e is the Euler’s ratio.. For more information visit my post on Euler’s ratio by clicking here…. In a natural logarithm we can always write ln instead of loge and therefore  loge n can be written as ln n 3. Loga mn is always equal to : n x loga m or simply nloga m 4. Logan m is always equal to : 1/n x loga m or simply 1/nloga m 5. Log a + Log b is always equal to : Log (a x b) [Here I have taken base as 10 but this is possible for any base n] The inverse is also true!! 6. Log a – Log b is equal to : Log (a/b) [Here I have taken base as 10 but it’s possible for any base] The inverse is also true.. These were some properties that we may require in our objective of determining the number of digits and if you want more details on logarithm with sums just comment it below!!

ANTILOG:

You must also know about antilogarithm though we may use it very less in this post… Literally you can get it that it’s just the opposite of logarithm.. If n = loga  m then Antiloga n = m or an = m which means that antilog of any number to any base is just the base raised to the power of that number..

CHARACTERISTIC AND MANTISSA OF LOGARITHMS:

This is the topic which you must learn carefully as all of the theory we are going to discuss is based on characteristic and mantissa of any log.. CHARACTERISTIC OF LOGARITHMS: Let there be two numbers a and b where a>1 and 0<b<1.. If there there n number of digits in a before the decimal point(if any) then the characteristic of that number a is (n-1). For example: The characteristic of 423.539 is 2 as the number of digits before decimal point is 3 and 3-1=2. If there are m number of zeroes after the decimal point till the first significant digit in b then the characteristic of b is defined as (m+1) and also is denoted by-(m+1). For example: The characteristic of 0.423539 is  as the number of zeroes between the decimal point and the first significant digit(4) is 0 and 0+1=1 which is denoted by -1. MANTISSA OF LOGARITHMS: Mantissa of any number is it’s property which is always present in the decimal portion of the common logarithm of that number. For finding mantissa we always use logarithmic tables!! USING MANTISSA AND CHARACTERISTIC IN A COMMON LOGARITHM: We already know that a common logarithm is the log of  number to the base of 10 and so we can easily find out common logarithms of any number if we know the characteristic and the mantissa of that number.. The characteristic is always the integral part and the mantissa is always the decimal part. Let’s do an example, Log 4567.. Now the characteristic of 4567 is 3 as the number of digits is 4 and 4-1=3. The mantissa of 4567,from the logarithmic table, is 0.6597. And so the Log 4567=3.6597  where the characteristic forms the part before decimal point and mantissa forms the part after decimal point!! It was so easy… wasn’t it?? Similarly Log 45.67 = 1.6597 The mantissa never changes and is fixed for a particular number!! Now You answer me, Log 0.04567= ?? The characteristic is –2 and mantissa is 6597. Therefore Log 0.04567 = -2.6597

CALCULATING NUMBER OF DIGITS IN nⁿ :

Now it will be very easy to explain our main objective… Recall my first question…

Calculate the number of digits in 2⁴³??

So let 2⁴³ be x. Therefore, 2⁴³ = x => Log 2⁴³ = Log x (We can take log of same bases on both side of an equation.) => 43 Log 2 = Log x ( See property no. 3) Now log 2 equals 0.3010 => 43 x 0.3010 = Log x => 12.943 = Log x Now we already know that the integral part of logarithm of any number to the base of 10 is the characteristic of that number!! Therefore the characteristic of x is 12 and characteristic is the number of digits – 1. Therefore the number of digits in x is 12 + 1= 13!! It was so easy and we needn’t count 2⁴³!! So test yourself now: What is the number of digits in 10099 ?? It is easier than the sum we did just before!! Try and comment your answer in the comment box!! So, Here was a quite useful and interesting post for you our readers!! -THANK YOU . .  

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