Given that ln(x)=log_exln(x)=logex
and are satisfied:
x>0,e>0,e!=1x>0,e>0,e≠1
We can apply the logarithmic properties:
ln(a*b)=ln(a)+ln(b)ln(a⋅b)=ln(a)+ln(b)
ln(a/b)=ln(a)-ln(b)ln(ab)=ln(a)−ln(b)
ln(a^b)=bln(a)ln(ab)=bln(a)
and remembering that:
a^(m/n)=root(n)(a^m)amn=n√am
a^-m=1/a^ma−m=1am
then:
ln(sqrt(3^-3*5^3*2^-2))=ln((3^-3*5^3*2^-2)^(1/2))=ln(√3−3⋅53⋅2−2)=ln((3−3⋅53⋅2−2)12)=
=1/2ln(3^-3*5^3*2^-2)=1/2(ln(3^-3)+ln(5^3)+ln(2^-2))==12ln(3−3⋅53⋅2−2)=12(ln(3−3)+ln(53)+ln(2−2))=
=1/2(-3ln(3)+3ln(5)-2ln(2))=12(−3ln(3)+3ln(5)−2ln(2))
Alternitevely:
ln((3^-3*5^3*2^-2)^(1/2))=1/2ln(3^-3*5^3*2^-2)=ln((3−3⋅53⋅2−2)12)=12ln(3−3⋅53⋅2−2)=
=1/2ln(5^3/(3^3*2^2))==12ln(5333⋅22)=
=1/2(ln(5^3)-ln(3^3)-ln(2^2))==12(ln(53)−ln(33)−ln(22))=
=1/2(3*ln(5)-3ln(3)-2ln(2))=12(3⋅ln(5)−3ln(3)−2ln(2))
