If ln(a) = 2, ln(b) = 3, an ln(c) = 5, evaluate the following:?

Question

(a) ln ( {a^{3} } / {b^{4}c^{3}) =

(b)ln sqrt{ b^{-4}c^{-4}a^{2}} =

(c) ln ( a^{-3}b^{-3} / {ln((bc)^{3})} =

d) (ln c^{-4}) ( ln {a}/{b^{3}})^{3} =

 

Answer 1

(a)

The Logarithmic values are

ln(a) = 2

ln(b) = 3

ln(c) = 5

= ln (a³) - ln(b^4*c³)                (Using Logarithmic Rule ln(a/b) = ln(a) - ln(b))

= ln (a³) - (ln(b^4) + ln(c³))     (Using Logarithmic Rule ln(ab) = ln(a) + ln(b))

= ln (a³) - ln(b^4) - ln(c³)

= 3 ln(a) - 4 ln(b) - 3 ln(c)      (Using Logarithmic Rule ln(a^n) = n*ln(a))

= 3*2 - 4*3 - 3*5

= 6 - 12 - 15

= -21

Therefore is equal to -21.

Answer 2

(b)

The Logarithmic values are

ln(a) = 2

ln(b) = 3

ln(c) = 5

   

          (Using Logarithmic Rule ln(a^n) = n*ln(a))

    (Using Logarithmic Rule ln(ab) = ln(a) + ln(b))

= -2 ln(b) - 2 ln(c) + ln(a)

= -2*3 - 2*5 + 2

= -14

Therefore is equal to -14. 

 

Answer 3

(c)

The Logarithmic values are

ln(a) = 2

ln(b) = 3

ln(c) = 5

                 (Using Logarithmic Rule ln(ab) = ln(a) + ln(b))

        (Using Logarithmic Rule ln(ab) = ln(a) + ln(b) and ln(a^n) = n*ln(a))             

        (Using Logarithmic Rule  ln(a^n) = n*ln(a))

Therefore is equal to -1/8.

Answer 4

(d)

The Logarithmic values are

ln(a) = 2

ln(b) = 3

ln(c) = 5

                  (Using Logarithmic Rule ln(a/b) = ln(a) - ln(b))

         (Using Logarithmic Rule ln(a^n) = n*ln(a))             

Therefore is equal to 6860.