LOGARITHM

WHAT IS A LOGARITHM?

A logarithm is the power to which a number must be raised in order to get some other number. For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:

log 100 = 2

because

10²

 = 100

This is an example of a base-ten logarithm. We call it a base ten logarithm because ten is the number that is raised to a power. The base unit is the number being raised to a power. There are logarithms using different base units. If you wanted, you could use two as a base unit. For instance, the base two logarithm of eight is three, because two raised to the power of three equals eight: 

log₂(8) = 3  because 2³ = 8

In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written  log

and a base ten logarithmic equation is usually written in the form: log a = r 

A natural logarithm is written ln

and a natural logarithmic equation is usually written in the form: ln a = r

So, when you see log by itself, it means base ten log. When you see ln, it means natural logarithm (we'll define natural logarithms below). In this course only base ten and natural logarithms will be used.

Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials.

In practical terms, I have found it useful to think of logs in terms of The Relationship:

y = bx

is equivalent to......

(means the exact same thing as)  

logb(y) = x

On the left-hand side above is the exponential statement "y = bx". On the right-hand side above, "logb(y) = x" is the equivalent logarithmic statement, which is pronounced "log-base-b of y equals x"; The value of the subscripted "b" is "the base of the logarithm", just as b is the base in the exponential expression "bx". And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations.

If you can remember this relationship (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms.

(I coined the term "The Relationship" myself. You will not find it in your text, and your teachers and tutors will have no idea what you're talking about if you mention it to them. "The Relationship" is entirely non-standard terminology. Why do I use it anyway? Because it works.)

By the way: If you noticed that I switched the variables between the two boxes displaying "The Relationship", you've got a sharp eye. I did that on purpose, to stress that the point is not the variables themselves, but how they move.

• 
  

  
  

  Convert "

  6

  ³ 
  = 216" to the equivalent logarithmic expression.

          To convert, the base (that is, the 6)remains the same, but the 3 and the 216                    switch sides. 

         This gives me:  log6(216) = 3   

• 
  

  Convert "

  log

  Log41024= 5" to the equivalent exponential expression.

To convert, the base (that is, the 4) remains the same, but the 1024 and the 5 switch sides. This gives me: 

4

⁵ = 1024

Prime Numbers

A Prime Number can be divided evenly only by 1 or itself. And it must be a whole number greater than 1.

Example: 2, 3, 5, 7, 11, etc.

Twin Primes

A pair of prime numbers that differ by 2 (successive odd numbers that are both Prime numbers).

Examples: (3,5), (5,7), (11,13), ...

It is not known whether the set of twin prime numbers ends or not.

Coprimes or Relatively prime numbers

A pair of numbers not having any common factors other than 1 or -1. (Or alternatively theirgreatest common factor is 1 or -1)

Example: 15 and 28 are co-prime, because the factors of 15 (1,3,5,15), and the factors of 28 (1,2,4,7,14,28) are not in common (except for 1).

Mersenne's Primes

Prime numbers of the form 2ⁿ-1 where n must itself be prime.

3, 7, 31, 127 etc. are Mersenne primes.

Not all such numbers are primes. For example, 2047 (i.e. 2¹¹-1) is not a prime number. It is divisible by 23 and 89.

Mersenne's Primes are named after the French monk, theologian, philosopher and number-theorist Marin Mersenne (1588-1648 AD).

Perfect numbers

Any positive integer that is equal to the sum of its distinct proper factors (factors other than the number itself).

Example: 6 (proper factors: 1,2,3) is a Perfect number because 1+2+3=6.

Example: 28 (proper factors: 1,2,4,7,14) is also a Perfect number, because 1+2+4+7+14=28.

Euclid proved that 2^n-1(2ⁿ-1) is an even perfect number when 2ⁿ-1 is a Mersenne prime. These are now called Euclid numbers and Euler proved that all even Perfect numbers are of this form for some positive prime number n. Thus, 6, 28, 496 are Perfect and correspond to values of 3, 7, and 31 for 2ⁿ-1 in the formula.

This table shows the results for n=1 to 13 which include the first five Perfect numbers:

n	2n-1	2n-1(2n-1)	Perfect?	Comment
1	1	1	No	n is not prime
2	3	6	Yes	n is prime, 2n-1 is prime
3	7	28	Yes	n is prime, 2n-1 is prime
4	15	120	No	n is not prime
5	31	496	Yes	n is prime, 2n-1 is prime
6	63	2016	No	n is not prime
7	127	8128	Yes	n is prime, 2n-1 is prime
8 to 10	...	...	No	not prime
11	2047	2096128	No	n is prime, but 2n-1 is not prime
12	4095	8386560	No	n is not prime
13	8191	33550336	Yes	n is prime, 2n-1 is prime

Whether there are infinitely many even Perfect numbers or any odd perfect numbers remain unsolved questions.

Abundant numbers

Any positive integer that is less than the sum of its distinct proper factors (factors other than the number itself).

Example: 12 is abundant since its distinct proper factors are 1, 2, 3, 4, and 6 whose sum is 16.

Deficient Numbers

Any positive integer that exceeds the sum of its distinct proper factors.

Any prime number is deficient, because it has only one proper factor: 1.

All numbers of the form 2ⁿ are also deficient.

Example: 32 (=2⁵) is a deficient number because the sum of its distinct proper factors is 31 (1+2+4+8+16).

Furthermore, numbers of the form pⁿ are always deficient where p is a prime number and n is a positive integer.

Example: 3⁵=243.

The factors of 243 other than itself are 81, 27, 9, 3 and 1.

Sum of these factors is 121, which is less than 243.

Similarly, 5⁶=15625, the factors of 15625 are 1,5,25,125,625, and 3125.

The sum of these is 3906 which is less than 15625.

Amicable numbers

A pair of integers, each of which is the sum of the distinct proper factors of the other.

Example: 220 and 284 are amicable numbers because:

• Factors of 220 (other than itself) are 1,2,4,5,10,11,20,22,44,55,110.
  Sum of these factors = 284
• Factors of 284 (other than itself) are 1,2,4,71,142.
  Sum of these factors = 220.

Euclid's proof that the set of prime numbers is endless

The proof works by showing that if we assume that there is a biggest prime number, then there is a contradiction.

We can number all the primes in ascending order, so that P1 = 2, P2 = 3, P3 = 5 and so on. If we assume that there are just n primes, then the biggest prime will be labelled Pn . Now we can form the number Q by multiplying together all these primes and adding 1, so

Q = (P1 × P2 × P3 × P4... × Pn) + 1

Now we can see that if we divide Q by any of our n primes there is always a remainder of 1, so Q is not divisible by any of the primes.

But we know that all positive integers are either primes or can be decomposed into a product of primes. This means that either Q must be prime or Q must be divisible by primes that are larger than Pn.

Our assumption that Pn is the biggest prime has led us to a contradiction, so this assumption must be false, so there is no biggest prime.

Goldbach's Conjecture

The conjecture that every even number (greater than or equal to 6) can be written as the sum of two odd prime numbers.

Goldbach's Conjecture is named after Prussian born number-theorist and analyst Christian Goldbach (1690-1764 AD) who was a Professor of Mathematics at, and the Historian of, the Russian Imperial Academy. He was also the tutor of Peter the Great, and was a member of the Tsar's foreign ministry.

Goldbach also conjectured that all odd numbers are the sum of three odd primes: Vinogradov's theorem shows this true of all except possibly finitely many odd numbers.

Definition 
Two numbers are said to be Coprimes when they have only 1 as a common factor. 

Following examples would further help you to understand Coprimes 

Example 1 = Are numbers 6 and 25 Coprime ? 
Answer = Two given numbers are 6 and 25 
Factors of 6 = 1, 2, 3, 6 
Factors of 25 = 1, 5, 25 
On comparing the factors of numbers 6 and 25, you can see that both have only 1 as a common factor,
Hence we can say that numbers 6 and 25 are Coprimes. 

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Example 2 = Are numbers 12 and 21 Coprime ? 
Answer = Two given numbers are 12 and 21 
Factors of 12 = 1, 2, 3, 4, 6, 12 
Factors of 21 = 1, 3, 7, 21 
On comparing the factors of numbers 12 and 21, you can see that both have 1 and 3 as common factors, 
Hence, we can say that numbers 12 and 21 are not Coprimes. 

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Example 3 = Are numbers 27 and 16 Coprime ? 
Answer = Two given numbers are 27 and 16 
Factors of 27 = 1, 3, 9, 27 
Factors of 16 = 1, 2, 4, 16 
On comparing the factors of numbers 27 and 16, you can see that both have 1 as a common factor, 
Hence we can say that numbers 27 and 16 are Coprimes.