SOLUTION: Write the expression as a single logarithm with a coefficient of 1.
logb(8) + 5[logb(7 + x) − 1/2 logb(6 − x)]
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Question 365314: Write the expression as a single logarithm with a coefficient of 1.
logb(8) + 5[logb(7 + x) − 1/2 logb(6 − x)]
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
There are three properties of logarithms we will need for this problem:
All of these properties can be used in both directions. But for this problem we will using them from left to right. The third property allows us to move a coefficient into the argument as an exponent. We will use it to move the (1/2) in front of the last log into its argument:
Since 1/2 as an exponent means square root, I'll replace the last argument with a square root:
Next, inside the parentheses we have two logarithms of the same base with coefficients of 1 and with a "-" between them. This is a job for the second property:
or
Next we can use the third property again to move the 5 from in front of the second log:
And last, with two logs of the same base with coefficients of 1 and a "+" between them, we can use the first property to combine them:
which is a single logarithm with a coefficient of 1. (NOTE: You do have a square root in a denominator. So you may need to rationalize the denominator. I'll leave that up to you.)
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