SOLUTION: Simplify. Log4 ((sqrt(x^5*y^7))/z*w^4)

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Question 832425: Simplify.
Log4 ((sqrt(x^5*y^7))/z*w^4)

Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--

THE PROBLEM:
Simplify:
log%284%2C%28%28sqrt%28x%5E5%2Ay%5E7%29%29%2Fz%2Aw%5E4%29%29

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%284%2C%28%28sqrt%28x%5E5%2Ay%5E7%29%29%2Aw%5E4%29%29-log%284%2Cz%29

Apply the Log Law for Multiplication-to-Addition: log(ab) = log(a) + log(b)
log%284%2C%28sqrt%28x%5E5%2Ay%5E7%29%29%29-log%284%2C%28z%29%29%2Blog%284%2C%28w%5E4%29%29

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%284%2C%28%28x%5E5%2Ay%5E7%29%5E0.5%29%29-log%284%2C%28z%29%29%2Blog%284%2C%28w%5E4%29%29

Apply the Power Rule for Logarithms: log(a^n) = n * log(a)
0.5%2Alog%284%2C%28x%5E5%2Ay%5E7%29%29-log%284%2C%28z%29%29%2B4%2Alog%284%2C%28w%29%29

Apply the Log Law for Multiplication-to-Addition again.


Apply the power rule again.


Distribute the 0.5 to clear some parentheses.




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