How do you expand $ln (sqrt((3^-2)(5^3)(2^-2)))$ ?

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Answer 1 · 2016-01-08T08:48:37

$ln(sqrt(3^-3*5^3*2^-2))=1/2(3ln(5)-2ln(2)-3ln(3))$

Given that $ln(x)=log_ex$
and are satisfied:
$x>0,e>0,e!=1$

We can apply the logarithmic properties:

$ln(a*b)=ln(a)+ln(b)$
$ln(a/b)=ln(a)-ln(b)$
$ln(a^b)=bln(a)$

and remembering that:

$a^(m/n)=root(n)(a^m)$
$a^-m=1/a^m$

then:

$ln(sqrt(3^-3*5^3*2^-2))=ln((3^-3*5^3*2^-2)^(1/2))=$
$=1/2ln(3^-3*5^3*2^-2)=1/2(ln(3^-3)+ln(5^3)+ln(2^-2))=$
$=1/2(-3ln(3)+3ln(5)-2ln(2))$

Alternitevely:

$ln((3^-3*5^3*2^-2)^(1/2))=1/2ln(3^-3*5^3*2^-2)=$
$=1/2ln(5^3/(3^3*2^2))=$
$=1/2(ln(5^3)-ln(3^3)-ln(2^2))=$
$=1/2(3*ln(5)-3ln(3)-2ln(2))$

How do you expand ln (sqrt((3^-2)(5^3)(2^-2)))ln (sqrt((3^-2)(5^3)(2^-2)))?