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 2² 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

Top
Previous Lesson - Properties of Mantissa in Logarithm