Logarithm

The logarithm is inverse function of Exponent, that is, raising a number to a power.

For  example,

Exponent function: 5 x 5 x 5 = 125, we get 125 when we multiply 5 with itself three times, mathematical expression of this is,  5³ = 125

The third power of 5 is 125, because 125 is the product of three factors of 5, then the logarithm of 125 with respect to base 5 is 3, mathematical expression of this is,   log5 125 = 3

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We can observe that,

  • The number which we are multiplying is the “base”.
  • No. of times we multiply the number or the power of a number or exponent of a number will the answer of the log function.

Clearly,

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For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000(1000 = 10 × 10 × 10 = 103); the multiplication is repeated three times. It can be written as, log10 1000 = 3 

Definition:

Logarithm is the power to which a number must be raised in order to get some other number.

The logarithm of a positive real number m with respect to base a, a positive real number not equal to 1, is the exponent by which a must be raised to yield x.

In other words, the logarithm of m to base a is the solution x to the equation, it is denoted by logm, and is pronounced as “the logarithm of m to base a” or “the base-a logarithm of m“.

In the equation,

ax = m, y = loga(m) , a > 0 and a≠ 1

LOG FACTS:

  1. br = a is the equivalent to logb a=r (This is the definition of a logarithm.)

⇒ 2³ = 8  is equivalent to log2 8=3

 

2.   log 0 is undefined.

⇒ because its is not possible get zero by raising any number to the power of any number. You can never reach zero, you can only approach it using an infinitely large and negative power

3.   log 1 = 0

The logarithm of 1 is always zero, no matter what the base of the logarithm is. This is because any number raised to 0 equals 1. i.e.,

10 = 1

20 = 1

30 = 1

n0 = 1

Therefore, ln 1 = 0 also.

4. loga a=1

Because a1 = a.

5. logamn = logam + logan

Let p = logam and q = logan

ap = m and aq = n

logamn = logaapaq

= logaap+q

= p +  q (since logaa = 1)

=  logam + logan

6. loga m/n = logam – logan

Let p = logam and q = logan

ap = m and aq = n

logam/n = logaap/logaaq

= logaap-q

= p –  q (since logaa = 1)

=  logam – logan

7. logamn = n logam

means that the logarithm of a number raised to some power, it is the same as multiplying the logarithm of that number by the value of the power.

For example:

log (32) = 2 * log 3 

2 * 0.477 = 0.954

8.  logb(m)=logc(m)/Logc(b)

Let logb(m)=x

bx=m

logc(bx)=logc(m) [logc(e)=logc(f).]

x•logc(b)=logc(m) [ logb(mx)=x•logb(m).]

x=logc(m)/logc(b)

logb(m)=logc(m)/logc(b)



 

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