```Question 202938
There are several properties of logarithms which are useful when you want to manipulate expressions involving them:<ol><li>{{{log(a, (x*y)) = log(a, (x)) + log(a, (y))}}}
Used from left to right, this property can be used to separate factors in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the sum of two logarithms into a single, equivalent logarithm.</li><li>{{{log(a, (x/y)) = log(a, (x)) - log(a, (y))}}}
Used from left to right, this property can be used to separate the numerator and denominator of a fraction in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the difference of two logarithms into a single, equivalent logarithm.</li><li>{{{log(a, (x^y)) = y*log(a, (x))}}}
Used from left to right, this property can be used to "move" of the argument of a logarithm out in front of the logarithm (as a coefficient. Used from right to left this can be used to "move" a coefficient of a logarithm into the arguments as the exponent of the logarithm.</li><li>{{{log(a, (x)) = (log(b, (x)))/(log(b, (a)))}}}
This property is used most used from left to right in order to change the base of a logarithm from "a" to "b".</li></ol>
Since we are interested in separating the x's, y's and z's into separate terms we will be using the first three properties from left to right.<br>
{{{log(a, ((x^4)/(yz^2)))}}}
Since the argument is a fraction, I'll use property #2 to split the fraction into separate logs:
{{{log(a, (x^4)) - log(a, (yz^2))}}}
Now I can move the exponent of the argument of the first log out in front using property #3:
{{{4*log(a, (x)) - log(a, (yz^2))}}}
Now I'll separate the product in the argument of the second log using property #1:
{{{4*log(a, (x)) - (log(a, (y)) + log(a, (z^2)))}}}
Note the parentheses around the new expression. This is <i>critical</i> since there is a subtraction in front! Next I'll "move" the exponent out the argument of the 3rd log using property #3:
{{{4*log(a, (x)) - (log(a, (y)) + 2*log(a, (z)))}}}
And finally I'll subtract the expression in the parentheses:
{{{4*log(a, (x)) - log(a, (y)) - 2*log(a, (z))}}}```