Logarithms

suppose there is a hen that lays an egg and hatches a hen out daily and the next day those two start doing the same thing that keeps going on and all are immortal. Just tell me how long will they take to reach 62 hens? How to calculate this? Can you find some pattern on which they are increasing in numbers? Let’s start from the beginning , a hen gives birth to another hen. the very next day both the hens give birth to two more hens and now they all are four hens in number. On the third day, those four hens are giving birth to four more hens and so on. Looks like they are doubling in number. And ultimately you can say that at the end of six days there will be more than 62 hens and on the fifth day, there will be 32 hens , hence 62 hens are resulted on the sixth day . as one can understand from the example that the rate of growth at which they are growing in number daily is 2.

 Day 1 : 🐔 🐣 

Day 2 : 🐔 🐣  🐔 🐣 

Day 3 : 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣

Day 4 : 🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣

Day 5 : 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣

Day 6 : 🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣  🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣 🐔 🐣  🐔 🐣

 There could be anything in place of hens and at any growth rate or decline rate , all you have to do is just find the term by which the  a particular change in something is happening if applicable.

Logarithms is similar to the concept Mentioned above . Logarithms Can tell us the time , term, attempts or something similar by which a thing has changed at a specific rate.

There are two most used bases  those are ‘10’ ie. Log10 or simply ‘lg’  and ‘e’ (Euler’s number, spoken as “oiler’s number”)  ie. Loge known as natural log or simply ‘ln’ . otherwise any kind of numerical value can be put as a base and no short symbol is used ie. Log 2 log3 log4 ………. (keeping the rules in mind which are mentioned below) :

So how do we write log of a given number?  

Suppose a² = b ( a raised to the power 2 or simply "a square" ) Exponent form, so we can write The syntax of it as 

 log_ab=2 

 For example, 

log₁₀10 = lg10 = 1 , this is the log form of 10¹ = 10

 log₁₀100= 2 Is From 10²=100

 and log₂8=3  Is logarithmic form of 2³=8

How to read the log function, log_ba. it reads,”Log to the base b of a” or “log base b of a”

Here, ‘b’ (base) can be any number positive number except 1 ( b>1)( generally  we are not allowed to have negatives as base)

For example :   if we have 1 as a base, it will not satisfy the  argument except 1

Log₁1=0 looks legit because  10=1  And 

Log₁1=1  As we know 1¹=1 

Except these two logarithmic functions,  base 1 is not working out at all

Log₁2= undefined 

since 1^x= 1  ≠2                 ( because whatever exponent you pick )

Similarly log₁3= undefined and so on 

‘a’ is called ‘argument’ of the log, this also can neither be negative nor zero.

For example:    Log_x0= undefined,  because 

Whatever base you pick and raise it to even 0  but the minimum value is 1

_X0=1        where x is a positive integer      

And the logarithmic value of the 'Argument' is called 'exponent'. And it could be zero or any negative value 

For example : log₁₀1=0 ,  from the exponent form 100=1

Log₁₀0.1=-1 , from the exponent form 10^-1 =0.1

The reason behind “no negative base or argument” is that you can not plot a graph which does not contradict itself. We will discuss it with a graph

From the graph, it is clear that the bigger the value of ‘x’ gets, the bigger the value of its log gets as well and if we continuously increase the value of x and put them all is the plane simultaneously, a smooth curve (without any discontinuity) is drawn. One more thing is noticeable that the bigger is base’s value, the smaller is the log of the same argument. 

But as I said  No negative base, no negative argument . From the graph as it can be seen that the values of log_-2X = y (for convenience in plotting on a 2 - axis plane)  

If we put 1 as argument of the logarithmic function

log_-21=0

Likewise, for x = 2, can be find the exponent of -2 that results in an argument +2? Generally,  we are not going to find it. And this is the same case for x =3 and  for many numbers so on.  OK,  we have an idea,  let's follow y = 0, 1, 2, 3 ...............

-2¹ = -2 ie. Log _-2-2= 1

-2² = +4 ie. Log_-24 = 2

-2³ = -8 ie. Log_-2-8= 3

-2⁴ =+16 ie. Log_-216 = 4 

..........................................

So you can see the coordinate dots with sequence numbers, they look random and not easy to draw a graph through these dots because we can not find the coordinates for y = 1.1 , 1.2  , 1.3 (any decimal number) in between the whole numbers. Looks unusual so we can't draw on at least a 2D plane. 

       I think now you have decent knowledge of  logarithmic functions. Ciao!