Definition: If $\displaystyle a$ is a positive number, other than $\displaystyle 1$ and $\displaystyle x$ is a rational number such that $\displaystyle a^x = n$, then we say that logarithm of $\displaystyle n$ to the base $\displaystyle a$ is $\displaystyle x$. It is written as $\displaystyle log_an = x$.

i.e. $\displaystyle a^x = n \Longleftrightarrow log_an = x$

Some examples for you to grasp this concept a little more: $\displaystyle 2^5 = 32 \Longleftrightarrow \log_2{32} = 5$ $\displaystyle 3^4 = 81 \Longleftrightarrow \log_3{81} = 4$ $\displaystyle 10^3 = 1000 \Longleftrightarrow \log_{10}1000 = 3$ $\displaystyle 81^{\frac{1}{4}} = 3 \Longleftrightarrow \log_{81}3 = \frac{1}{4}$ $\displaystyle 7^0 = 1 \Longleftrightarrow \log_71 = 0$ $\displaystyle 10^{-2} = 0.01 \Longleftrightarrow \log_10{0.01} = -2$

Note: For any positive real number a, we know that $\displaystyle a^0 = 1$. Hence $\displaystyle \log_a1 = 0$ and $\displaystyle \log_aa = 1$ and

Fundamental Laws of Logarithms:

First Law:

If $\displaystyle m, \ n$ are positive rational numbers, then $\displaystyle \log_a(mn) = \log_a m + \log_a n$ i.e. the logarithms of the product of two numbers is equal to the sum of their logarithms.

We can use this law and extrapolate it to multiple situations like:

• If $\displaystyle m, n, p$ are positive numbers, then $\displaystyle \log_a(mnp) = \log_a m + \log_a n + \log_a p$
• If $\displaystyle x_1, x_2, x_3, ... , x_n$ are positive numbers, then $\displaystyle \log_a(x_1.x_2.x_3...x_n) = \log_a x_1 + \log_a x_2 + \log_a x_3 \cdots + \log_a x_n$
• If $\displaystyle x, y$ are real numbers such that $\displaystyle xy>0$, then $\displaystyle \log_a(xy) = \log_a |x| + \log_a |y|$

Example 1: Prove $\displaystyle \log_{10} (1+2+3) = \log_{10} 1 + \log_{10} 2 + \log_{10} 3$

LHS $\displaystyle = \log_{10} (1+2+3) = \log_{10} (6) = \log_{10} (1\times 2 \times 3)$ $\displaystyle = \log_{10} 1 + \log_{10} 2 + \log_{10} 3$ = RHS

Example 2: Prove $\displaystyle \log_{10} 2 + 1 = \log_{10} 20$

LHS $\displaystyle = \log_{10} 2 + 1$ $\displaystyle = \log_{10} 2 + \log_{10} 10 = \log_{10} (2 \times 10) = \log_{10} 20 =$ RHS

Second Law:

If $\displaystyle m, \ n$ are positive rational numbers, then $\displaystyle \log_a ( \frac{m}{n} ) = \log_a m - \log_a n$ i.e. the logarithms of the ratio of two numbers is equal to the difference of their logarithms.

The above law can also be stated as follows: If $\displaystyle x, y$ are real numbers such that $\displaystyle xy>0 \text{ , then } \log_a ( \frac{x}{y} ) = \log_a |x| - \log_a |y|$

Example 1: Evaluate: $\displaystyle \log_{10} 500 - \log_{10} 5 = \log_{10} ( \frac{500}{5} ) = \log_{10} 100 = 2 \log_{10} 10 = 2$

Example 2: Evaluate: $\displaystyle \log_6 72 - log_6 2 = \log_6 ( \frac{72}{2} ) = \log_6 36 = 6 \log_6 6 = 6$

Third Law:

If $\displaystyle m, \ n$ are positive rational numbers, then $\displaystyle \log_a(m^n) = n. \log_a m$ $\displaystyle \text{Example 1: } \log_2(3^2) = 2. \log_2 3$ $\displaystyle \text{Example 2: } \log_3(2^2) = 2. \log_3 2$

Fourth law: $\displaystyle \log_a 1 = 0$ i.e. the log of $\displaystyle 1$ to any base is always $\displaystyle 0$.

Fifth law: $\displaystyle \log_a a = 1$ i.e. the log of any positive number to the same base is always $\displaystyle 1$.

Example 1: Evaluate: $\displaystyle \frac{\log_a 125}{\log_a \sqrt{5}} = \frac{\log_a 5^3}{\log_a 5^{1/2}} = \frac{3 \log_a 5}{\frac{1}{2} \log_a 5} = 6$

Example 2: Evaluate: $\displaystyle \log_2 ( \log_2 ( \log_2 16)) = \log_2 ( \log_2 ( \log_2 2^4))$ $\displaystyle = \log_2 ( \log_2 4) = \log_2 ( \log_2 2^2) = \log_2 2 = 1$

Sixth law (Base change formula):

If $\displaystyle m$ is a positive rational number and $\displaystyle a, b$ are positive real numbers such that $\displaystyle a \neq 1 \text{ and } b \neq 1 \text{ , then } \log_a m = \frac{\log_b m}{\log_b a}$ .

In the above formula if b = m, then we get $\displaystyle \log_a m = \frac{\log_m m}{\log_m a} = \log_a m . \log_m a = 1$

Example 1: Evaluate: $\displaystyle \log_b a . \log_c b . \log_a c = (\log_b a. \log_c b). \log_a c = \log_c a . \log_a c = 1$

Since $\displaystyle \log_b a = \frac{\log_c a}{\log_c b}$  we get

Example 2: Evaluate: $\displaystyle \frac{1}{\log_2 42} + \frac{1}{\log_3 42} + \frac{1}{\log_7 42}$ $\displaystyle = \log_{42} 2 + \log_{42} 3 + \log_{42} 7$ $\displaystyle = \log_{42} (2 \times 3 \times 7) = \log_{42} 42 = 1$

Seventh Law:

If $\displaystyle a$ is a positive real number and $\displaystyle n$ is a positive rational number, then $\displaystyle a^{\log_a n} = n$ $\displaystyle \text{Example 1: } 3^{\log_3 8} = 8$ $\displaystyle \text{Example 2: } 2^{3\log_2 5} = 2^{\log_2 5^3} = 5^3$

Eighth Law:

If $\displaystyle a$ is a positive real number and $\displaystyle n$ is a positive rational number, then $\displaystyle \log_{a^n} n^p = \frac{p}{q} \log_a n$ $\displaystyle \text{Example 1: } \log_{81} 243 = \log_{3^4} 3^5 = \frac{5}{4} \log_3 3 = \frac{5}{4}$ $\displaystyle \text{Example 2: } \log_{1024} 64 = \log_{2^{10}} 2^6 = \frac{6}{10} \log_2 2 = \frac{3}{5}$

Ninth Law:

If $\displaystyle x, y$ and $\displaystyle a(\neq 1)$ are positive real numbers, then $\displaystyle x^{\log_a y} = y^{\log_a x}$ $\displaystyle \text{Example 1: } 16^{\log 3} = (4^2)^{\log 3} = 4^{\log 3^2}= 4^{\log 9} = 9^{\log 4}$ $\displaystyle \text{Example 2: } 27^{\log 2} = (3^3)^{\log 2} = 3^{\log 2^3} = 3^{\log 8} = 8^{\log 3}$

As you would have noticed, we have listed all the key concepts of logarithms (as related to Class 9 students). We have also presented a couple of example / applications of the laws. However, we would recommend the student to solved all the problems as stated in Exercise 8(a). More you practice, the better.