SOLUTION: Given that log (base 7)x^2 = p and log (base 7) xy^2 = q, find log (base 7) (cube root over xy) in terms of p and q. hope you'll get my question right!

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Question 824341: Given that log (base 7)x^2 = p and log (base 7) xy^2 = q, find log (base 7) (cube root over xy) in terms of p and q.
hope you'll get my question right!
Found 2 solutions by Edwin McCravy, jsmallt9:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Given that and , find in terms of p and q.
        
        
         
             
                    
                    
                    
                    

We want to find:

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 =
 =
 =
 =
 =
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Edwin
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
and
What we are going to do is...
  1. Solve the above equations to get expressions for and .
  2. Rewrite in terms of and .
  3. Substitute in the expressions for and from step 1 into the expression we got in step 2.
  4. Simplify.
Solving for :
Using a property of logarithms, , to move the exponent out in front:

Dividing both sides by 2:


Solving for :
Using another property of logarithms, , we can split the argument into separate logs:

Substituting in the expression we got for :

Using the property to move the exponent:

Multiplying each side by 1/2:

Subtracting p/4:


Rewriting in terms of and :
Since a cube root is the same as an exponent of 1/3:

Moving the exponent:

Split the argument:

Substituting in our expressions for the two logs:

Simplifying...
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