Question 1103106
The hardest to understand (and remember) properties of logarithms are
flipping base and number: {{{log(Y,B)}}}{{{"="}}}{{{1/log(B,Y)}}}
and
change of base: {{{log(Y,X)}}}{{{"="}}}{{{log(B,X)/log(B,Y)}}} .
If you are going to "apply properties of logarithms" you need those two here.
You also need to know that {{{log(B,XY)=log(B,X)+log(B,Y)}}} and that
{{{log(B,X^n)=n*log(B,X)}}} .
 
1) If you do not want to (or cannot) memorize or remember properties, you do not need to.
It all flows logically from the meaning of logarithm.
{{{log(a,x)=p}}} means that {{{a^p=x}}} , and
{{{log(a,y)=q}}} means that {{{a^q=y}}} .
That means that {{{a^(p+q)=a^p*a^q=xy}}} ,
and if {{{xy=a^(p+q)}}} .
Then, you understand that
{{{xy^"1 / ( p+q )"}}}{{{"="}}}{{{(a^(p+q))^(1/(p+q))}}}{{{"="}}}{{{a^"(p+q) / (p+q)"}}}{{{"="}}}{{{a^1=a}}} ,
and the fact that {{{xy^"1 / ( p+q )"=a}}} means that
{{{log(xy,a)}}}{{{"="}}}{{{highlight(1/(p+q))}}} .
 
INVOKING PROPERTIES OF LOGARITHMS:
1)
{{{log(xy,a)}}}{{{"="}}}{{{1/log(a,xy)}}}{{{"="}}}{{{1/(log(a,x)+log(a,y))}}}{{{"="}}}{{{highlight(1/(p+q))}}} .
 
2)
{{{log(y,ax^2y^3)}}}{{{"="}}}{{{log(y,a)}}}{{{"+"}}}{{{log(y,x^2)}}}{{{"+"}}}{{{log(y,y^3)}}}{{{"="}}}{{{log(y,a)}}}{{{"+"}}}{{{2log(y,x)}}}{{{"+"}}}{{{3log(y,y)}}}{{{"="}}}{{{1/log(a,y)}}}{{{"+"}}}{{{2log(a,x)/log(a,y)}}}{{{"+"}}}{{{3*1}}}{{{"="}}}{{{highlight(1/q+2p/q+3)}}}

1) If you cannot memorize or remember properties, you do not need to.
It all flows logically from the meaning of logarithm.
{{{log(a,x)=p}}} means that {{{a^p=x}}} , and
{{{log(a,y)=q}}} means that {{{a^q=y}}} .
That means that {{{a^(p+q)=a^p*a^q=xy}}} ,
and if {{{xy=a^(p+q)}}} .
Then, you understand that
{{{xy^"1 / ( p+q )"}}}{{{"="}}}{{{(a^(p+q))^(1/(p+q))}}}{{{"="}}}{{{a^"(p+q) / (p+q)"}}}{{{"="}}}{{{a^1=a}}} ,
and the fact that {{{xy^"1 / ( p+q )"=a}}} means that
{{{log(xy,a)}}}{{{"="}}}{{{highlight(1/(p+q))}}} .

NOTE:
The "properties of logarithms" sound like something to memorize,
and something to be anxious about,
but there is no need to be scared.
{{{log(a,x)=p}}} is read as
the logarithm on base{{{a}}} of {{{x}}} is {{{p}}} .
The exponent on base {{{a}}} to get {{{x}}} is {{{p}}} sounds easier,
by just replacing the word "exponent" for the scary word "logarithm."