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We are given these two logarithms. Another base 12 logarithm we can use without having to be told is:
The "trick" to this kind of problem is to express the argument of the desired logarithm in terms of products, quotients and/or powers of the arguments of the known logarithms. In this problem the argument of the desired logarithm is 48. The arguments of the known logarithms are 3, 7 and 12. So we want to express 48 as some product(s), quotient(s) and/or power(s) of 3, 7 and/or 12.
First we can find that 48=12*4. But we don't know what is. Can we express a 4 in terms of 3, 7 and/or 12? Answer: 4 = 12/3. So now we have
Now we can find :
Now we can use a property of logarithms, , to split the logarithm of a product into the sum of the logarithms of the factors:
Next we can use another property of logarithms, , to split the logarithm of a quotient into the difference of the logarithms of the numerator and denominator:
We can now replace each logarithm with its known value:
And simplify:
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