document.write( "Question 202923: Write as a single logarith,and, if possible, simplify: log(a)(a/sqrt(x)-log(a)(sqrt(ax)) \n" ); document.write( "
Algebra.Com's Answer #153095 by jsmallt9(3758)\"\" \"About 
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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\"
    \n" ); document.write( "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\"
    \n" ); document.write( "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\"
    \n" ); document.write( "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\"
    \n" ); document.write( "This property is used most used from left to right in order to change the base of a logarithm from \"a\" to \"b\".

\n" ); document.write( "We will use these properties to simplify
\n" ); document.write( "\"log%28a%2C%28a%2Fsqrt%28x%29%29%29+-+log%28a%2C+sqrt%28ax%29%29\"
\n" ); document.write( "(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\".)
\n" ); document.write( "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:
\n" ); document.write( "\"log%28a%2C+%28a%29%29+-+log%28a%2C+%28sqrt%28x%29%29%29+-+log%28a%2C+%28sqrt%28ax%29%29%29\"
\n" ); document.write( "\"log%28a%2C+%28a%29%29+=+1\".
\n" ); document.write( "And square roots are the same as an exponent of (1/2). Using these facts to rewrite our expression we get:
\n" ); document.write( "\"1+-+log%28a%2C+%28x%5E%281%2F2%29%29%29+-+log%28a%2C+%28%28ax%29%5E%281%2F2%29%29%29\"
\n" ); document.write( "Now we can use property #3 above to \"move\" the exponents out from the arguments to the front of the 2ns and 3rd logs:
\n" ); document.write( "\"1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29log%28a%2C+%28ax%29%29\"
\n" ); document.write( "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.)
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\n" ); document.write( "Again \"log%28a%2C+%28a%29%29+=+1\".
\n" ); document.write( "\"1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29%281++%2B+log%28a%2C+%28x%29%29%29\"
\n" ); document.write( "Simplifying we get:
\n" ); document.write( "\"1+-+%281%2F2%29log%28a%2C+%28x%29%29+-+%281%2F2%29+-+%281%2F2%29log%28a%2C+%28x%29%29\"
\n" ); document.write( "Combining like terms (1 - (1/2) = 1/2; -1/2log - 1/2log = -1log):
\n" ); document.write( "\"1%2F2+-+log%28a%2C+%28x%29%29\"\r
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