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

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

Some examples for you to grasp this concept a little more:

$2^5 = 32 \Longleftrightarrow \log_2{32} = 5$

$3^4 = 81 \Longleftrightarrow \log_3{81} = 4$

$10^3 = 1000 \Longleftrightarrow \log_{10}1000 = 3$

$81^{\frac{1}{4}} = 3 \Longleftrightarrow \log_{81}3 = \frac{1}{4}$

$7^0 = 1 \Longleftrightarrow \log_71 = 0$

$10^{-2} = 0.01 \Longleftrightarrow \log_10{0.01} = -2$

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

Fundamental Laws of Logarithms:

First Law:

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

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

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

$= \log_{10} 1 + \log_{10} 2 + \log_{10} 3$ = RHS

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

LHS $= \log_{10} 2 + 1$

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

Second Law:

If $m, \ n$ are positive rational numbers, then  $\log_a ($ $\frac{m}{n}$ $) = \log_a m - \log_a n$ 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 $x, y$ are real numbers such that $xy>0$, then $\log_a ($ $\frac{x}{y}$ $) = \log_a |x| - \log_a |y|$

Example 1: Evaluate:

$\log_{10} 500 - \log_{10} 5 = \log_{10} ($ $\frac{500}{5}$ $) = \log_{10} 100 = 2 \log_{10} 10 = 2$

Example 2: Evaluate:

$\log_6 72 - log_6 2 = \log_6 ($ $\frac{72}{2}$ $) = \log_6 36 = 6 \log_6 6 = 6$

Third Law:

If $m, \ n$ are positive rational numbers, then $\log_a(m^n) = n. \log_a m$

Example 1: $\log_2(3^2) = 2. \log_2 3$

Example 2: $\log_3(2^2) = 2. \log_3 2$

Fourth law:

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

Fifth law:

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

Example 1: Evaluate:

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

Example 2: Evaluate:

$\log_2 ( \log_2 ( \log_2 16)) = \log_2 ( \log_2 ( \log_2 2^4))$

$= \log_2 ( \log_2 4) = \log_2 ( \log_2 2^2) = \log_2 2 = 1$

Sixth law (Base change formula):

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

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

Example 1: Evaluate:

$\log_b a . \log_c b . \log_a c = (\log_b a. \log_c b). \log_a c = \log_c a . \log_a c = 1$

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

Example 2: Evaluate:

$\frac{1}{\log_2 42}$ $+$ $\frac{1}{\log_3 42}$ $+$ $\frac{1}{\log_7 42}$

$= \log_{42} 2 + \log_{42} 3 + \log_{42} 7$

$= \log_{42} (2 \times 3 \times 7) = \log_{42} 42 = 1$

Seventh Law:

If $a$ is a positive real number and $n$ is a positive rational number, then $a^{\log_a n} = n$

Example 1: $3^{\log_3 8} = 8$

Example 2: $2^{3\log_2 5} = 2^{\log_2 5^3} = 5^3$

Eighth Law:

If $a$ is a positive real number and  $n$ is a positive rational number, then $\log_{a^n} n^p =$ $\frac{p}{q}$ $\log_a n$

Example 1: $\log_{81} 243 = \log_{3^4} 3^5 =$ $\frac{5}{4}$ $\log_3 3 =$ $\frac{5}{4}$

Example 2: $\log_{1024} 64 = \log_{2^{10}} 2^6 =$ $\frac{6}{10}$ $\log_2 2 =$ $\frac{3}{5}$

Ninth Law:

If $x, y$ and $a(\neq 1)$ are positive real numbers, then $x^{\log_a y} = y^{\log_a x}$

Example 1: $16^{\log 3} = (4^2)^{\log 3} = 4^{\log 3^2}= 4^{\log 9} = 9^{\log 4}$

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.