# Given ln 2=x and ln5=y, use the properties of logarithms to write ln square root 20 in terms of x and y.

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We have to write ln [sqrt 20] in terms of x = ln 2 and and y = ln 5

ln [sqrt 20]

=> ln [20^(1/2)]

use the property ln a^x = x*ln a

=> (1/2)*ln[ 20]

=> (1/2)*ln[2^2*5]

use the property ln(a*b)= ln a + ln b

=> (1/2)[ln(2^2) + ln 5]

=> (1/2)[2*ln 2 + ln 5]

=> (1/2)[2x + y]

**ln(sqrt 20) in terms of x and y is (1/2)*[2x + y]**

ln (sqrt20) = ln [(20)^(1/2)]

We'll use the power property of logarithms:

ln [(20)^(1/2)] = (ln 20)/2

ln 20 = ln (4*5)

We'll use the product property of logarithms:

ln(a*b) = ln a + ln b

ln (4*5) = ln 4 + ln 5

But ln 4 = ln (2^2)

ln 4 = 2*ln 2

ln 20 = 2*ln 2 + ln 5

ln 20 = 2x + y

ln (sqrt20) = (2x+y)/2

**The given logarithm written in terms of x and y is: ln (sqrt20) = (2x+y)/2.**

hi,

[sqrt 20] in terms of x = ln 2 and and y = ln 5

ln [sqrt 20]= ln [20^(1/2)]

by:- a^x = x*ln a

= (1/2)*ln[ 20]

= (1/2)*ln[(2^2)*5]

by mulitplication method ln(a*b)= ln a + ln b

= (1/2)[ln(2^2) + ln 5]

=> (1/2)[2*ln 2 + ln 5]

=> (1/2)[2x + y]

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