SOLUTION: 7 log6(x−4)+2log6(x+3)−1/2log6 x
write as the loarithm of a single expression
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Question 823741: 7 log6(x−4)+2log6(x+3)−1/2log6 x
write as the loarithm of a single expression
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
If these were like terms then we could just add or subtract them to combine them into a single logarithm. (Like logarithmic terms have the same bases and the same arguments.) These, however, are not like terms (same bases, yes, but all three arguments are different: x-4, x+3 and x.)
Fortunately there are three properties of logarithms which provide an laternate way to combine logarithmic terms:The first two can be used to combine logarithmic terms. (The first one for when the terms have a "+: between them and the second for when there is a "-".) They require that the logs have the same bases and that they have a coefficient of 1.
Our terms all have coefficients that are not 1's. But that is where the third property comes in. It can be used to "move" a coefficient into the argument as its exponent. So that is where we will start: Using the third property to move the coefficients out of the way:
Since 1/2 as an exponent is the same as a square root and since fractional exponents do not always display well on algebra.com, I am going to replace the fractional exponent with a square root before proceeding. (This is not something you need to do.)
Now we will use the first property to combine the first two terms:
And now we use the second property to combine the remaining terms:
This may be an acceptable answer. But we may want to rationalize the denominator:
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