SOLUTION: express as a single logarithm log a 7 + log a 12 choose from one of these answers log a 84 log 2a 19 log 2a 84 log a 19

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Question 268185: express as a single logarithm
log a 7 + log a 12
choose from one of these answers
log a 84
log 2a 19
log 2a 84
log a 19
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29

Logarithms can only be added if the bases and the arguments are the same. Your logarithms have the same base, a, but the arguments are different. So we cannot add them.

So how can log%28a%2C+%287%29%29+%2B+log%28a%2C+%2812%29%29 be expressed as a single logarithm like all the answers? Well, we have a property of logarithms which allows us to combine two logarithms which have a addition symbol between them: log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29. This property only requires that the bases are the same (and the logarithms have coefficients of 1). Using this on your expression we get:
log%28a%2C+%287%2A12%29%29
which simplifies to
log%28a%2C+%2884%29%29

When the bases and arguments are the same we can add (or subtract) logarithms. But when we do so, only the coefficients change. After the addition (or subtraction) the bases and arguments of the logarithm are still the same! For example:
log%28a%2C+%2815%29%29+%2B+3log%28a%2C+%2815%29%29+=+4log%28a%2C+%2815%29%29
It's just like
x + 3x = 4x.