SOLUTION: Write as a single logarith,and, if possible, simplify: log(a)(a/sqrt(x)-log(a)(sqrt(ax))

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Question 202923: Write as a single logarith,and, if possible, simplify: log(a)(a/sqrt(x)-log(a)(sqrt(ax))
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are several properties of logarithms which are useful when you want to manipulate expressions involving them:
  1. log%28a%2C+%28x%2Ay%29%29+=+log%28a%2C+%28x%29%29+%2B+log%28a%2C+%28y%29%29
    Used from left to right, this property can be used to separate factors in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the sum of two logarithms into a single, equivalent logarithm.
  2. log%28a%2C+%28x%2Fy%29%29+=+log%28a%2C+%28x%29%29+-+log%28a%2C+%28y%29%29
    Used from left to right, this property can be used to separate the numerator and denominator of a fraction in the argument of a logarithm into separate logarithms. Used from right to left this can be used to combine the difference of two logarithms into a single, equivalent logarithm.
  3. log%28a%2C+%28x%5Ey%29%29+=+y%2Alog%28a%2C+%28x%29%29
    Used from left to right, this property can be used to "move" of the argument of a logarithm out in front of the logarithm (as a coefficient. Used from right to left this can be used to "move" a coefficient of a logarithm into the arguments as the exponent of the logarithm.
  4. log%28a%2C+%28x%29%29+=+%28log%28b%2C+%28x%29%29%29%2F%28log%28b%2C+%28a%29%29%29
    This property is used most used from left to right in order to change the base of a logarithm from "a" to "b".

We will use these properties to simplify
log%28a%2C%28a%2Fsqrt%28x%29%29%29+-+log%28a%2C+sqrt%28ax%29%29
(If the above is not a correct representation of your problem, please repost the question using more parentheses to clarify "what goes with what and where".)
The argument of the first log is a fraction and we will use property #2 above to separate the numerator and denominator into separate logs:
log%28a%2C+%28a%29%29+-+log%28a%2C+%28sqrt%28x%29%29%29+-+log%28a%2C+%28sqrt%28ax%29%29%29
log%28a%2C+%28a%29%29+=+1.
And square roots are the same as an exponent of (1/2). Using these facts to rewrite our expression we get:
1+-+log%28a%2C+%28x%5E%281%2F2%29%29%29+-+log%28a%2C+%28%28ax%29%5E%281%2F2%29%29%29
Now we can use property #3 above to "move" the exponents out from the arguments to the front of the 2ns and 3rd logs:
1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29log%28a%2C+%28ax%29%29
Now we can use property #1 above to separate the "a" and the "x" in the third log into separate logs. (Note the parentheses used when substituting. They are critical.)

Again log%28a%2C+%28a%29%29+=+1.
1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29%281++%2B+log%28a%2C+%28x%29%29%29
Simplifying we get:
1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29+-+%281%2F2%29log%28a%2C+%28x%29%29
Combining like terms (1 - (1/2) = 1/2; -1/2log - 1/2log = -1log):
1%2F2+-+log%28a%2C+%28x%29%29