SOLUTION: Express as an equivalent expression, using the individual logarithms of w,x,y, and z {log(baseb)w^2x}/{y^3z}

Question 234917: Express as an equivalent expression, using the individual logarithms of w,x,y, and z

{log(baseb)w^2x}/{y^3z}
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
With "regular" Algebra you cannot modify an argument to a function. But for logarithms there are some properties we can use to do so:

This allows us rewrite the log of a product as the sum of the logs of the factors.
This allows us to rewrite the log of a quotient as the difference of the log of the numerator and denominator.
This allows us to "move" the exponent of the argument in a log to the front, as a coefficient.

Your problem is to take the log of a complex expression and, using the properties above, break it down into an expression of logs of simple expressions.

To start with the argument is a big fraction. We will use the second property above to "split" this into two logs:

Now we have two logs. And each one is a product. So we will use the first property above to split each of them. (Note the use of parentheses in making these substitutions. This is an excellent habit when subsituting part of an expression.)

which simplifies to:

(Note that with out the use of parentheses, we would not have a minus sign in front of . And without this minus sign we would no longer have a correct expression! This illustrates why it is always a good idea to use parentheses when making substitutions.)
And finally, we can use the third property above to "move" the exponents out of the arguments of the first and third logs:

And we now have rewritten the single log of a complex argument as an expression of "simple" logs.
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