An expression like a x a x a x a can be written as x4.
x4 is read as x raised to power 4.
in the expression x4 , 4 is the index and x is the base.
In other words, an index is the number of times the base is multiplied by itself.
The laws of indices
Multiplication law
when the numbers with the same base are multiplied together, the result is same as raised the base of one with the sum of all their indices.
For example:
a2 X a5 x a7 = a(2+5+7) = a13
Conclusion:
An index of a product is the sum of the given indices provided the bases are the same.
in general case :am x an = am+n
Division law
When numbers of the same base are divided, the result is like raising the common base with the difference of their indices.
prove: let a10 ÷ a7 =
=a x a x a = a3 = a(10-7)
conclusion:
Index of a quotient from two numbers with the same bases is given by the index of the divisor subtracted from the index of the dividend.
in general case: am ÷ an = am-n
power rule
consider the case (b3)3 = b3 x b3 x b3 = b9 = b3×3
conclusion: when a number is raised to a certain power is raised to power, the result is like multiplying the indices together.
in general : (am)n = am*n
Zero Index rule
a0 = 1
generally:
any number raised to power zero is equal to 1
example: (100000000)0 = 1
Negative indices
it states :
Prove:
conclusion:
number raised to negative indices is same as reciprocal of the same number raised to positive power
Fractional indices
in other words, nth root of a number is like raised that number with a reciprocal of the number.
The laws of Logarithm
- log(ab) = log a + log b; in other words log of product of a and b is like summation of their individual logarithm
- log(a/b) = log a – log b; that is, the log of quotient of a divided by b is like subtracting log b from log a
- n x log a = log an; that is product of n and log a is equal to log a raised to power of n.
Example
Evaluate without using mathematical tables or a calculator:
2 log 5 – 1/2 log16+2 log 40
solution
using the rule n x log a = log an ; we rewrite the expression as:
log 52 – log 16(1/2) + log 402 = log 25 – log 4 + log 1600
rem:
The above expression can be simplified from the expression:
log(a/b) = log a – log b meaning log 25-log4 = log(25/4) and hence we rewrite the expression as:
and from the rule: log(ab) = log a + log b; we have
log (25/4 x 1600 ) = log(25 x 400) = log 10000
and log 10000 = log 104 = 4 log 10
but log 10 = 1
hence 4 x 1 = 4
Hence the whole expression evaluate evaluates to 4.
Example
Given that log 2 = 0.3010 and log 3 = 0.4771 evaluate:
(a) log 6 (b) log 1.5 ( c) log 54
solution
general expression: log (ab) = log a + log b
(a)
log 6 = log (3 x 2) = log 3 + log 2
= 0.4771 + 0.3010 = 0.7781
(b)
log 1.5 = log(3/2) = log 3 – log 2
= 0.4771 – 0.3010 = 0.1761
( c) log 54 = log (27 x 2)=log (33 x 2)
log 33 + log 2 = 3 log 3 + log 2
3(0.4771) + 0.3010
= 1.7323