SOLUTION: Express as a sum, difference, or multiple of logarithms. log(subscript 2)(square root of x / a^2)

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Question 198679: Express as a sum, difference, or multiple of logarithms.
log(subscript 2)(square root of x / a^2)
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2C%28sqrt%28x%29%2Fa%5E2%29%29 Start with the given expression.


log%282%2C%28sqrt%28x%29%29%29-log%282%2C%28a%5E2%29%29 Break up the log using the identity log%28b%2C%28A%2FB%29%29=log%28b%2C%28A%29%29-log%28b%2C%28B%29%29


log%282%2C%28x%5E%281%2F2%29%29%29-log%282%2C%28a%5E2%29%29 Convert to exponential notation


%281%2F2%29log%282%2C%28x%29%29-2log%282%2C%28a%29%29 Pull down the exponents using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


So log%282%2C%28sqrt%28x%29%2Fa%5E2%29%29=%281%2F2%29log%282%2C%28x%29%29-2log%282%2C%28a%29%29

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2C+sqrt%28x%2Fa%5E2%29%29
The general rule is;
log%28a%2Fb%29+=+log%28a%29+-+log%28b%29
log%282%2C%28sqrt%28x%29+%2F+sqrt%28a%5E2%29%29%29
log%282%2C+x%5E%281%2F2%29%29+-+log%282%2C+%28a%5E2%29%5E+%281%2F2%29%29
Another general rule is
log%28a%5Eb%29+=+b%2Alog%28a%29
%281%2F2%29%2Alog%282%2Cx%29+-+log%282%2Ca%29 answer
check answer:
Say x+=+1 and a+=+4
log%282%2C+sqrt%28x%2Fa%5E2%29%29
log%282%2C+sqrt%281%2F16%29%29
log%282%2C+1%2F4%29+=+-2
and
%281%2F2%29%2Alog%282%2Cx%29+-+log%282%2Ca%29
%281%2F2%29%2Alog%282%2C1%29+-+log%282%2C4%29
%281%2F2%29%2A0+-+2+=+-2
OK
Note that log%282%2C-a%29 is impossible. There is no
log to a positive base that will result in a negative
number.