SOLUTION: if log a=2 and log b=3, what is the numerical value of log square root a / b^3 ?

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Question 389942: if log a=2 and log b=3, what is the numerical value of
log square root a / b^3 ?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28sqrt%28a%29%2Fb%5E3%29%29
Since you're given log(a) and log(b), this expression can be evaluated if we can rewrite this expression in terms of log(a) and/or log(b). To do this we start by using the property of logarithms, log%28z%2C+%28p%2Fq%29%29+=+log%28x%2C+%28p%29%29+-+log%28x%2C+%28q%29%29, to write the logarithm of a quotient as the difference of the logarithms of the numerator and denominator:
log%28%28sqrt%28a%29%29%29+-+log%28%28b%5E3%29%29
Since square roots are the same as an exponent of 1/2 this can be written as:
log%28%28a%5E%281%2F2%29%29%29+-+log%28%28b%5E3%29%29
Next we can use another property of logarithms, log%28x%2C+%28p%5Eq%29%29+=+q%2Alog%28x%2C+%28p%29%29, to move the exponents in the arguments out in front:
%281%2F2%29log%28%28a%29%29+-+3log%28%28b%29%29
We now have the expression in terms of log(a) and log(b). We can substitute in their values:
%281%2F2%29%282%29+-+3%283%29
which simplifies as follows:
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