Chapter 5 Indices And Logarithms

(~) = square root
(*) = divide


1. a^n = a x a x a x a ....................... x a

a is the base and n is the index.

2 (a) Zero index, a^0 = 1
(b) Negative integer index, a^(-n) = 1/(a^n)
(c) Fractional index, a^(1/n) = n~(a)
a^(m/n) = (n~(a^m)) =(n~(a))^m

3. Laws of indeces:
(a) a^m x a^n = a^(m + n)
(b) a^m * a^n = a^(m - n)
(c) (a^m)^n = a^(m x n)

4. If y = a^x, then log(a)y = x.
(a) If a^0 = 1 therefore log(a)1 = 0
(b) If a^1 = a therefore log(a) a = 1

5. Laws of logarithms:
(a) log (a) xy = log (a) x + log (a) y
(b) log (a) (x/y) = log (a) x - log (a) y
(c) log (a) x^n = n log (a) x

6. Changing the base of logarithms:
(a) log (a) b = (log (c) b) / (log (c) a)
(b) log (a) b = 1/(log(b) a) , when c = b


Example:

a) 27^x / 81^ 2x = 1/ 243
3^3x / (3^4)^2x = 1/3^5
3^(3x-8x) = 3^(-5)
-5x = -5
x = 1

b) log (10)(x - 5) = log (10) (x - 1) + 2
log (10) (x-5) - log (10) (x - 1) = 2
log (10) (x - 5/x - 1) = 2
(x - 5/x - 1) = 10^2
x - 5 = 100 (x - 1)
x - 5 = 100 x - 100
99 x = 95
x = 95/99