"Logarithms, Surds and Indices" is one of the easiest topics in the quantitative aptitude section of the SSC exam. Although the number of formulae is high, the basic concepts are very simple to understand and apply. There are no shortcuts to remember and the scope of the questions that can be asked is very limited. The accuracy of answering questions from this section is very high and good students tend to score very well here.
Power:
Let a be a real number and n be a positive number.
$$a^n = a\times a\times a\times…..\times a$$
Here $$a^n$$ is called ‘a to the power of n’.
Here, a is called the base while n is called the power.
Laws of indices:
Surd:
Let ‘a’ be a real number and ‘n’ be a positive integer.
$$\sqrt[n]{a} = a^{1/n}$$
Here, $$\sqrt[n]{a}$$ is called ‘$$n^{th}$$ root of a’.
Quadratic surd:
A surd of order 2 is called a quadratic surd.
Eg: $$\sqrt{2}$$
Cubic surd:
A surd of order 3 is called a cubic surd.
Eg: $$\sqrt[3]{2}$$
Q) Find the value of $$\sqrt{12+\sqrt{12+\sqrt{12+....\infty}}}$$.
Solution:
$$12 = 3\times4$$ which is in the form of n(n+1).
Hence, $$\sqrt{12+\sqrt{12+\sqrt{12+....\infty}}} = 4$$
Q) Find the value of $$\sqrt{12-\sqrt{12-\sqrt{12-....\infty}}}$$.
Solution:
$$12 = 3\times4$$ which is in the form of n(n+1).
Hence, $$\sqrt{12-\sqrt{12-\sqrt{12-....\infty}}} = 3$$
Logarithm:
Let ‘a’ and ‘m’ be positive real numbers and $$a^x = m$$, then ‘x’ is called logarithm of ‘m’ to the base ‘a’.
$$a^x = m$$ can be written as $$x = log_a m$$
The logarithm of a number consists of two parts - an integral part, which is called characteristic and a decimal part, which is called the mantissa.
Example: In log 415 = 2.618, the characteristic is 2 and the mantissa is 0.618
Note: Mantissa is always positive. If the logarithmic value is negative, 1 should be added to the mantissa and subtracted from the characteristic so as to make the mantissa positive
Example: log 0.2 = -0.6989.
-0.6989 = (-1-0)+(1-0.6989) = -1+0.3011
Hence, The characteristic is $$\bar{1}$$ and mantissa is 0.3011.
