Indices, Surds and Logarithms

Example 4Solve each of the following.(a) 3 x − 1 + 3 x = 12 (b) 2 x + 2 x + 3 = 72 (c) 4 x + 1 + 2 2 x = 20

Example 3 (Solving index equation simultaneously)Solve the following simultaneous equations. 2 x .4 2 y = 8 3 x 9 y = 1 27

Example 2Solve each of the following.(a) 27 ( 81 3 x ) = 1 (b) 81 n + 2 = 1 3 n 27 n − 1 (c) 8 x − 1 = 4 2 x + 3

METHOD: Comparison of indices or base If  the base are the same , when a x = a y , then x = y If  the index are the same , when a x = b x , then a = b Using common logarithm (If base and index are NOT the same) a x … Read more

Example 3Given that log 3 x = b  , express log x 9 x in term of b. Example 4Given that log 2 3 = 1.585   and log 2 5 = 2.322  , evaluate the following. (a) log 8 15 (b) log 5 0.6 (c) log 15 30 (d) log 16 45 Solution:

Change of Base of Logarithms     log a b= log c b log c a  and    log a b= 1 log b a   Example: Find the value of the following: a. log 25 100 b. log 3 0.45 Answer: (a)   log 25 100= log 10 100 log 10 25 = log 10 10 … Read more

Example 3 Given that log 7 4 = 0.712   and log 7 5 = 0.827  , evaluate the following. (a) log 7 20 (b) log 7 1 1 4 (c) log 7 0.8 (d) log 7 28 (e) log 7 140 (f) log 7 100 (g) log 7 0.25 (h) log 7 35 64

Example 2 Find the value of the following. (a) log 2 7 + log 2 12 − log 2 21 (b) 3 log 10 5 + 2 log 10 2 − log 10 5 (c) 2 log 10 3 − log 10 3 + log 10 3 1 3 (d) log 3 3 p + … Read more

4.2a Laws of Logarithms   Law 1:   log a x y = log a x + log a y   Example:   log 5 25 x = log 5 25 + log 5 x   Beware!!   log a x + log a y ≠ log a ( x + y )      Law 2:   log a ( x y ) … Read more

Example 2Solve the following equations:(a) log 3 5 x = 2 (b) log x + 1 81 = 2 (c) log x 5 x − 6 = 2