Question 832425
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Hi, there--

THE PROBLEM:
Simplify:
{{{log(4,((sqrt(x^5*y^7))/z*w^4))}}}

A SOLUTION:
We work from the outside in, using the log laws.

Apply the Log Law for Division-to-Subtraction: log(a/b) = log(a) - log(b)
{{{log(4,((sqrt(x^5*y^7))*w^4))-log(4,z)}}}

Apply the Log Law for Multiplication-to-Addition: log(ab) = log(a) + log(b)
{{{log(4,(sqrt(x^5*y^7)))-log(4,(z))+log(4,(w^4))}}}

Now, we are ready to deal with the square root. Recall that take the square root of a number is equivalent 
to raising that number to the 1/2=0.5 power.
{{{log(4,((x^5*y^7)^0.5))-log(4,(z))+log(4,(w^4))}}}

Apply the Power Rule for Logarithms: log(a^n) = n * log(a)
{{{0.5*log(4,(x^5*y^7))-log(4,(z))+4*log(4,(w))}}}

Apply the Log Law for Multiplication-to-Addition again.
{{{0.5*(log(4,(x^5))+log(4,(y^7)))-log(4,(z))+4*log(4,(w))}}}

Apply the power rule again.
{{{0.5*(5*log(4,(x))+7*log(4,(y)))-log(4,(z))+4*log(4,(w))}}}

Distribute the 0.5 to clear some parentheses.
{{{2.5*log(4,(x))+3.5*log(4,(y))-log(4,(z))+4*log(4,(w))}}}



Hope this helps! Feel free to email if you have any questions about the solution.


Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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