Logarithms and Anti-Logarithms

Logarithms (and Antilogarithms) are primarily used to make calculations of powers and roots of any number easier, although they have numerous other real world applications.

Logarithms

Logarithm of a number N with respect another number B (known as base) is the power to which the base B must be raised to get the number N. Let's suppose 3 positive real numbers (i.e. numbers greater than 0) B, P and N such that,

BP= N (That is B to the power P is equal to N)

Therefore by definition of word logarithm we have

logBN = P
\(\Rightarrow\) B\(^{(\log_B N)}\)= N

Where logBN denotes logarithm of number N to the base B. Two important observations related to this formula are:

  1. logB1=0 (That is log of 1 to any base is always 0)
  2. logBB=1 (That is log of a number with same base is equal to 1)
Though logarithm of any number can be calculated with respect to any base, most common numbers that are used as base for calculation related to logarithms are:
  1. Base 10: Such logarithms are known as Common Logarithms.
  2. Base 'e': Such logarthims are known as Natural Logarithms.
  3. Base 2: Such logarithms are known as Binary Logarithms.

Rules of Logarithms:


There are 4 primary rules that apply to logarithms irrespective of the base used to calculate the logarithm.
  1. Production Addition Rule: This rule states that the logarithm of product of any numbers is equal to the sum of logarithms of those numbers. That is,

    logk (A x B x C x D... x N) = logkA + logkB + logkC + logkD +... logkN

    Proof: We will prove the rule for the following simple case:

    log k(A x B) = log k A + log k B
    Let log k A = T
    \(\Rightarrow\) A=kT
    Now, Let log k B = S
    \(\Rightarrow\)B =kS

    Therefore
    A x B = kT x kS
    \(\Rightarrow\)A x B = kT+S
    \(\Rightarrow\)log k (A x B) = T+S (By Definition of Logarithm)
    \(\Rightarrow\)log k (A x B) = log kA + log kB

    An interesting rule that can be derived through this rule itself is the following

    logkAN = N log kA

    Proof:

    log k AN = log k (A x A x A x.... N Times A) = log k A + log k A + log k A +.... N Times log k A = N x log k A

  2. Division Subtraction Rule: This rule states that the logarithm of result of division two of any two numbers is equal to the difference of logarithms of the dividend and the divisor. That is,

    log k (A/B) = log k A - log k B

    Proof:

    Let log k A = T
    \(\Rightarrow\) A=kT
    Now, Let log k B = S
    \(\Rightarrow\)B =kS

    Therefore
    A/B = kT / kS
    \(\Rightarrow\) A/B = kT-S
    \(\Rightarrow\)log k (A/B) = T-S (By Definition of Logarithm)
    \(\Rightarrow\)log k (A/B) = log k A - log k B

  3. Reciprocal Rule: This rule states that the logarithm of number N to the base B is reciprocal of logarithm of number B to the base N. That is,

    log B N = 1 / log N B

    Proof:

    Let log B N = P
    \(\Rightarrow\) BP = N

    Taking log to the base N on both sides

    log N BP= log N N
    \(\Rightarrow\)P x log N B = 1
    \(\Rightarrow\)log N B = 1/P
    \(\Rightarrow\)log NB = 1 / log B N

  4. Chain Rule: This rule can be best stated with the following formula,

    log A B x log B C = log A C

    Proof:

    Let log A B = T
    \(\Rightarrow\) B = AT
    Now Let log B C = N
    \(\Rightarrow\)C = BN
    \(\Rightarrow\)C = (AT) N

    Taking log to the base A on both sides

    log A C = log A (AT) N
    \(\Rightarrow\)log A C = log A (A) TN
    \(\Rightarrow\)log A C = T x N x log A (A)
    \(\Rightarrow\)log A C = T x N x 1
    \(\Rightarrow\)log A C = log A B x log B C

The Reciprocal Rule and the Chain Rule can be together used to find out the logarithm of any number to any base. For eg. Suppose we are given a table of values for log to the base 10 (Common logarithms) and we have to find out

log X Y
\(\Rightarrow\)log X 10 x log 10 Y
\(\Rightarrow\)log 10 Y / log 10 X

Since both log 10 X and log 10 Y can be found out from the table we can find out log X Y.

Anti-Logarithms

Anti-Logarithms provide the actual number whose logarithmic value for a particular base is given. Antilog of Log of a number gives back the number. That is

Antilog B(logBN) = N

Because of this, if Log and Antilog values are known for any particular base, then it can be used to calculate any power (or root) of any number. For example, Suppose we have to calculate

N = TS

Here we are given the value of T and S and we have to find the value of N. Taking log10 on both sides we get

log10N = log10(TS)
\(\Rightarrow\)log10N = S x log10 T

Taking Antilog10 on both sides we have

Antilog 10(log10N) = Antilog 10(S x log10 T)
\(\Rightarrow\)N = Antilog 10(S x log10 T)

By calculating Antilog 10(S x log10 T) we can find the value of N.