SOLUTION: Simplify using at most a single logarithm: logA^(3)-logB^(2/3)+logA^(1/3)+logB^(5/3)

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Simplify using at most a single logarithm: logA^(3)-logB^(2/3)+logA^(1/3)+logB^(5/3)      Log On


   

Question 335748: Simplify using at most a single logarithm:
logA^(3)-logB^(2/3)+logA^(1/3)+logB^(5/3)
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
I believe this means:



If so, then this is the same as:

log%28%28%28a%5E3+%2A+a%5E%281%2F3%29+%2A+b%5E%285%2F3%29%29+%2F+b%282%2F3%29%29%29

You can simplify this expression to be equal to:

log%28%28a%5E%2810%2F3%29+%2A+b%29%29

You can confirm this to be true by plugging any values for a and b and seeing if both equations provide you with the same answer.

For example, letting a = 1 and b = 2 yields the following:

log%28%28a%5E%2810%2F3%29+%2A+b%29%29 = log%28%281%5E%2810%2F3%29+%2A+2%29%29 which equals log%28%282%29%29 which equals .301029996

and the following:

which equals which equals which equals 0+-+.200686664+%2B+0+%2B+.501716659 which equals .301029996

Since you get the same answer with both equations, then the simplification is confirmed as good.

You used the following concepts to solve this.

log(x) + log(y) = log(x*y)

log(x) - log(y) = log(x/y)

In particular:

log%28%28a%5E3%29%29+%2B+log%28%28a%5E%281%2F3%29%29%29+%2B+log%28%28b%5E%285%2F3%29%29%29 = log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29

Also in particular:

log%28%28a%5E3%2Aa%281%2F3%29%2Ab%285%2F3%29%29%29+-+log%28%28b%5E%282%2F3%29%29%29 = log%28%28%28a%5E3%2Aa%5E%281%2F3%29%2Ab%285%2F3%29%29%2Fb%282%2F3%29%29%29%29

Some more concepts:

x%5Em+%2A+x%5En+=+x%5E%28m%2Bn%29

x%5Em+%2F+x%5En+=+x%5E%28m-n%29

In particular:

a%5E%283%29+%2A+a%5E%281%2F3%29+=+a%5E%283+%2B+1%2F3%29+=+a%5E%2810%2F3%29

Also in particular: