Question 211997
express as (all three) a sum, difference and product of logarithms, without using exponents
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The 3 manipulative rules of logarithms which you need are

(a)    {{{log(B,AC)=log(B,A)+log(B,C)}}}

(b)    {{{log(B,A/C)=log(B,A)-log(B,C)}}}

(c)    {{{log(B,A^C)=C*log(B,A)}}}

You will also need the rule

(d)    {{{sqrt(A)=A^(1/2)}}}



{{{log(b,sqrt(x^3y^2/z^9))}}}

First we use rule (d) and rewrite the above as

{{{log(b,(x^3y^2/z^9)^(1/2))}}} 

Next we use rule (c)

{{{(1/2)log(b,(x^3y^2/z^9))}}}

Next we use rule (b)

{{{(1/2)(log(b,x^3y^2)-log(b,z^9))}}}

Next we use rule (a)

{{{(1/2)(log(b,x^3)+log(b,y^2)-log(b,z^9))}}}

Next we use rule (c) on all three terms in the
parentheses:

{{{(1/2)(3log(b,x)+2log(b,y)-9log(b,z))}}}

Distribute the {{{(1/2)}}}

{{{(3/2)log(b,x)+log(b,y)-(9/2)log(b,z)}}}

Edwin</pre>