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Having been given the log(a) and log(b), we will be able to find the numerical value of the above if we can express in terms of log(a) and/or log(b).
To change this given logarithm into an expression of log(a) and/or log(b) we will be using the properties of logarithms which allow us to rewrite a logarithm as an expression of other logarithms with other arguments:
Since we have a somewhat complicated argument and we want simple arguments of just "a" or "b", we will be using the properties from right to left, from the more complicated arguments to the simpler ones.
Since our argument is a quotient, we will start by using the second property. This property tells us that the log of a quotient is equal to the difference of the logs of the numerator and denominator:
None of the properties indicate what can be done with a square root. But, if you remember fractional exponents, square roots are powers of 1/2. So we can rewrite the first argument with an exponent of 1/2:
Now we can use the third property which tells us that the log of a power is equal to the exponent times the log of the base:
We now have an expression involving log(a) and log(b). We can now substitute in the given values for these logs:
which simplifies as follows:
1 - 9
-8
So if log(a) = 2 and log(b) = 3.
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