## Solved Examples in Logarithms

#### Algebra > Logarithms > Solved Examples

13.Solved Examples in Logarithms: Now let us solve a few number of problems on logarithms to apply all of the formulas and concepts learned in this lesson:
1.Solve the following for x
1. log 10[ (log 3 (log 4 64)]
2. log 5 (log 6 36) = log x 4
Solution1: log 4 64 = log 4 43
= 3 log 4 4         {from Law 3. log a (p)n = n log a p }
= 3.1= 3         { log 4 4 = 1, since log of every number to base itself is 1}
Set log 4 64 = 3 in the given question, we get
log 10[ (log 3 3)]
= log 10 1         { log33 = 1}
= 0         { because log of 1 to any base is 0}

Solution2: log 4 (log 6 36) = log x 4
log 6 36
= log 6 6²        { log a (p)n = n log a p}
= 2log 6 6        { log 66=1, since log of every number to base itself is 1}
=2
set 2 in log 6 36 in the question, we get
log 4 (log 6 36) = log 4 2
set this value in the question and we have
log 4 2 = log x 4……………………….(1)
now log 42 = log 2
= ½ (log 22) = ½ (1) = ½
Put this value ½ in (1), we get
½ = log x 4
x1/2 = 4
squaring of both sides, we get
x = 4² = 16
2. Solve log 2 (-p2 +10p) = 4
Solution: From log definition, we have if log a x = n, then an = x
so, if log 2 (-p2 +10p) = 4, then
-p2 +10p = 24
-p2 +10p = 16
p2 -10p +16 = 0
we now have a quadratic equation in p.
we must solve this quadratic equation in p with the method of factorization p2 - 8p – 2p + 16 = 0
p(p – 8 )- 2 (p – 8) = 0
(p – 2)(p – 8) = 0
So, p – 2 = 0 or p – 8 = 0
So, p = 2 or p = 8