How do you write this as a single logarithm? log(a)U - log(a)V + 2log(a)W. The answer is log(a) (UW^2/V)

Question

Can someone please break down the steps for me?

Answer 1

When we shrink a logarithmic statement like this into one expression there are a couple things to remember

1. Subtraction of logarithms is the same as dividing
2. log x - log y is equivalent to log ^x/_y
3. Addition is the same as multiplication
4. log x + log y is equivalent to log (xy)
5. Multiplying is the same as raising to a power
6. X log y is equivalent to log y^x

So to compress this into one logarithmic statement we can start by combining the subtraction part:

log_a U - log_a V + 2 log_a W = log_a (^U/_V) + 2 log_a W

Now we can compress the 2 log_a W part

log_a (^U/_V) + 2 log_a W = log_a (^U/_V) + log_a W²

And to finish we can write the U and W² part as multiplication

log_a (^U/_V) + log_a W² = log_a (UW²/V)