Logarithmic differentiation issue Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. 1/x(x+1)(x+2) and I know that log((ab)/c)=log(a)+log(b)−log(c)

Winston Todd 2022-10-22 Answered
Logarithmic differentiation issue
Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on.
1 / x ( x + 1 ) ( x + 2 )
and I know that
l o g ( ( a b ) / c ) = l o g ( a ) + l o g ( b ) l o g ( c )
So I tried to use that rule here and did:
l n ( a ) l n ( b ) l n ( c )
and got:
l n ( 1 ) l n ( x ( x + 1 ) l n ( x ( x + 2 )<>rbwhich simplifies to:
0 l n ( x 2 + x ) l n ( x 2 + 2 x )
and then I look at the solution which gives:
y = ( 1 / ( x ( x + 1 ) ( x + 2 ) ) ) ( 1 / x + 1 / ( x + 1 ) + 1 / ( x + 2 ) )
I'm just kind of confused on what I am doing wrong.
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Answers (1)

Besagnoe9
Answered 2022-10-23 Author has 9 answers
You have that
ln f ( x ) = ln 1 x ( x + 1 ) ( x + 2 ) = ln 1 l n x ln ( x + 1 ) ln ( x + 2 )
= l n x ln ( x + 1 ) ln ( x + 2 ) .
Taking derivatives:
f ( x ) f ( x ) = 1 x 1 x + 1 1 x + 2 .
Thus
f ( x ) = 1 x ( x + 1 ) ( x + 2 ) ( 1 x + 1 x + 1 + 1 x + 2 ) .
Note a difference in sign with your expected answer.
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