Contract the expressions. that is, use the properties of logarithms

Question

to write each expression as a single logarithm with a coefficient of 1.?

A). ln 3-2 ln 7+ ln 14

B). ln 3-2 ln(7+8)

C). ln 3-2(ln 4+ ln 8)

Answer 1

(A).

The expressin is ln 3 - 2 ln 7 + ln 14.

Apply power property of logarithm : .

ln 3 - 2 ln 7 + ln 14 = ln 3 - ln 7² + ln 14

= ln 3 - ln 49 + ln 14

Apply quotient property of logarithm : .

= ln (3/49) + ln 14

Apply product property of logarithm : .

= ln [ (3/49) * 14 ]

= ln [6/7].

Therefore, ln 3 - 2 ln 7 + ln 14 = ln [6/7].

(B).

The expressin is ln 3 - 2 ln (7 + 8).

Apply power property of logarithm : .

ln 3 - 2 ln (7 + 8) = ln 3 - ln (7 + 8)²

= ln 3 - ln (15)²

Apply quotient property of logarithm : .

= ln (3/15²)

= ln [3/(15*15)]

= ln [1/75].

Therefore, ln 3 - 2 ln (7 + 8) = ln [1/75].

(C).

The expressin is ln 3 - 2(ln 4 + ln 8).

Apply product property of logarithm : .

ln 3 - 2(ln 4 + ln 8) = ln 3 - 2ln (4*8)

= ln 3 - 2ln 32

Apply power property of logarithm : .

= ln 3 - ln 32²

Apply quotient property of logarithm : .

= ln (3/32²)

= ln [3/1024].

Therefore, ln 3 - 2(ln 4 + ln 8) = ln [3/1024].