Logarithms and Antilogarithms are used to Calculate of Powers and Roots of Numbers .
Logarithm (or Log) 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 \(N\). For example,
given 3 Numbers \(B\), \(P\) and \(N\) such that,
\(B^P=N\)
then by definition Logarithm we have
\(\log_B N = P\)
Where \(\log_B N = P\) denotes Logarithm of Number N to the Base \(B\). Hence, Calculating Logarithms is basically Finding out the Power when a Base and a Number are given.
Following gives the Properties of the Logarithms
Logarithms are Not Defined if value of the Base is 0 or 1. That is
\(\log_0 N = \log_1 N=\) Not Defined
Logarithms are Not Defined for value 0. That is
\(\log_B 0 =\) Not Defined
Log of 1 to Any Base is Always 0. That is
\(\log_B 1 = 0\)
Log of a Number with Same Base is equal to 1. That is
\(\log_B B = 1\)
The value of the Logarithm are Real Numbers only when Value of Both Base and the Number for which Logarithm is calculated are Real Numbers Greater than 0. Otherwise, the value of the Logarithm are Complex Numbers.
The value of the Logarithm are Positive Real Numbers when Either Both Value of Base and the Number for which Logarithm is calculated are Real Numbers Greater than 1 Or Both are Real Numbers Between 0 and 1.
The value of the Logarithm are Negative Real Numbers when Value of Only One of either the Base Or the Number for which Logarithm is calculated are Real Numbers Between 0 and 1.
Although Logarithm of Any Number can be calculated with respect to Any Base, the most common Numbers that are used as Base for calculation related to Logarithms are
Base 10: Such logarithms are known as Common Logarithms.
Base 'e': Such logarthims are known as Natural Logarithms.
Base 2: Such logarithms are known as Binary Logarithms.
The following are 5 Rules of Logarithms irrespective of the Base used for calculation
Power-Multiplication Rule: This rule states that the Calculating Logarithm of a Number raised to the Power of another Number is same as Calculating the Logarithm of that Number Multiplied by that Power. That is
\(\log_B A^N = N \log_B A\)
Following gives the Derivation of this Rule
Let \(\log_B A^N=P\)
\(\Rightarrow A^N=B^P\) ...(1)
Now, let \(\log_B A=Q\)
\(\Rightarrow A=B^Q\)
\(\Rightarrow A^N=({B^Q})^N=B^{NQ}\) ...(2)
From equation (1) and (2) we have
\(B^P=B^{NQ}\)
\(\Rightarrow P=NQ\)
Therefore \(\log_B A^N = N \log_B A\)
Product-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,
Division-Subtraction Rule: This rule states that the Logarithm of Result of Division Any 2 Numbers is equal to the Difference of Logarithms of the Dividend and the Divisor. That is,
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,
\(\log_B N= \frac{1}{\log_N B}\)
Following gives the Derivation of this Rule
Let \(\log_B N = P\hspace{.5cm}\Rightarrow N=B^P\) ...(7)
Let \(\log_N B = Q\hspace{.5cm}\Rightarrow B=N^Q\) ...(8)
From equation (7) and (8) we have
\(N^1=B^P= {(N^Q)}^P= N^{P \times Q}\)
\(\Rightarrow P \times Q=1\)
\(\Rightarrow \log_B N \times \log_N B=1\)
\(\Rightarrow \log_B N= \frac{1}{\log_N B}\)
Chain Rule: This rule is given by the following formula,
\(\log_A B \times \log_B C = \log_A C\)
Following gives the Derivation of this Rule
Let \(\log_A B = P\hspace{.5cm}\Rightarrow B=A^P\) ...(9)
Let \(\log_B C = Q\hspace{.5cm}\Rightarrow C=B^Q\) ...(10)
From equation (9) and (10) we have
\(C=B^Q={(A^P)}^Q\)
\(\Rightarrow C=A^{P \times Q}\)
Taking \(\log_A\) on both sides we have
\(\log_A C=log_A A^{P \times Q}\)
From Power-Multiplication Rule we have
\(\log_A C=(P \times Q) \times log_A A\)
Since \(\log_A A=1\) therefore
\(\log_A C=P \times Q\)
\(\Rightarrow \log_A C=\log_A B \times \log_B C\)
The Reciprocal Rule and the Chain Rule can be together used to find out the Logarithm of Any Number to Any Base. For example, suppose we are given a table of values for Log to the Base 10 (Common logarithms)
and we have to find out \(\log_X Y\), it can be done using following
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 provide the Actual Number whose Logarithmic Value is given for a Particular Base. Antilog of the Log of a Number gives back the Number. That is
\(AntiLog_B (\log_B N) = N\) ...(11)
But we know that if
\(\log_B N = P\hspace{.5cm}\Rightarrow B^P= N\hspace{.5cm}\Rightarrow B^{(\log_B N)}= N\) ...(12)
Hence, Calculating Anti-Logarithm is basically Finding out the resultant Number when a Base and a Power are given by raising the Base to the Power..
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
\(N = T^S\)
Here we are given the value of \(T\) and \(S\) and we have to find the value of \(N\). Taking \(\log_{10}\) on both sides we get
\(\log_{10} N= \log_{10} T^S\)
From Power-Multiplication Rule we get
\(\log_{10} N= S \times \log_{10} T\)
Taking \(AntiLog_{10}\) on both sides we have
\(AntiLog_{10} (\log_{10} N)= 10^{(\log_{10} N)}=AntiLog_{10} (S \times \log_{10} T)=10^{ (S \times \log_{10} T)}\)
\(\Rightarrow N=10^{ (S \times \log_{10} T)}\)
Hence, by calculating \(10^{ (S \times \log_{10} T)}\) we can find the value of \(N\).