Solving Logarithms problem ( Add Maths)

One of the most important ability you need to master when working on a logarithm problem is to be able to see the relationship between a logarithm and indices. Beside that, you will need to familiarise yourself with the law of logarithms. Otherwise when you see a problem, it will take more time than necessary to finish a log problem.

Let's have a look at one log problem:

Question #1:

log5 x   = 1 + 2 logx 5

Find value of x

Solution:

log5 x   = 1 + 2 logx 5 

log5 x   = 1 + 2/log5 x    ( Note: change of base in formula : loga b = 1/logb a )

(log5 x)^2 =  log5 x + 2     (Note:  multiply both side with log5 x to eliminate the denominator)

(log5 x)^2 - log5 x - 2 = 0   ( Note: move everything to left hand side)

(log5 x - 2) ( log5 x + 1) = 0   ( Note: factorise)

When log5 x = 2
                 x = 5^2
                  x=25

When log5 x = -1
                 x = 5^-1
                 x = 1/5

Hence, x = 1/5 and 25

If you are not sure about your answers, replace the value of x with 1/5 and 25 in the equation and see if they fit in.



log5 25   = 1 + 2 log25 5
        2    = 1 + 2(0.5)
              = 1 + 1


log5 1/5   = 1 + 2 log1/5 5
        -1    = 1 + 2(-1)
                = 1 - 2 
                = -1

Question #2 : 

log5 pq = 2 + 3 log5 p - log5 q

Express p in term of q

 

Solution: The steps below are elaborated to allow for detailed explanations - refer the note on each respective lines. Some obvious lines can be omitted in your answer.

log5 pq = 2 + 3 log5 p - log5 q

log5 p + log5 q = log5 5^2 + 3 log5p - log5 q      ( Note 1: use the log law loga MN = loga M + loga N) 
                                                                                  (Note 2: Remember: loga M^b = b loga M)
log5 p - 3 log5 p = log5 5^2 - log5 q - log5 q    (Note 3: move all p to the left side and all q to the right side.)
  -2 log5 p = log5 5^2 -  2log5 q 

2log5 p = -log5 5^2 + 2 log5 q                            ( Note 4 : change negative to positive)

2 log5 p =  - 2 log5 5 + 2 log5 q

log5 p =  - log5 5 + log5 q                                   ( Note 5: divide with 2)

lo5 5 p = log5 (q/5)

       p  = q/5