SOLUTION: Express as single logarithm and simply ln (9sec theta) + ln(9 cos theta)

Question 391908: Express as single logarithm and simply
ln (9sec theta) + ln(9 cos theta)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Actual addition of logarithms requires that the two logarithms have the same bases and the same arguments. When this is so you just add the coefficients (just like when you add variables). But your logarithms do not have the same arguments.

On the other hand there is a property of logarithms, , which allows us to combine two logarithms when

the bases are the same.
there is a "+" between them.
both coefficients are 1's

Your logarithms fit the requirements of this property so we can combine them into one according to the pattern of the property. IOW, the two logarithms can be replaced by a single logarithm of the product of the arguments:
ln((9*sec(theta))*(9*cos(theta)))
The argument is all multiplication so we can use the Commutative and Associative Properties to rearrange the order and grouping in any way we choose. Multiplying the sec and cos first we get a 1 (as long as sec is defined) since cos and sec are reciprocals of each other. So this becomes:
ln(9*9*1)
or
ln(81)
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