SOLUTION: write as the sum or difference of two or more logarithms: 1.log(4x) 2.ln 15 3.log 3/xy

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Question 268905: write as the sum or difference of two or more logarithms:
1.log(4x)
2.ln 15
3.log 3/xy
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are many possible answers to these problems. The purpose of these problems, I think, is to get you to show that you know how to use the properties of logarithms. So I will solve the problems this way.

The two properties of logarithms that we will need are:
  • log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29
    If the argument of a logarithm is or can written as a product (multiplication), then this property let's you rewrite it as the sum of the logarithms of the factors.
  • log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29
    If the argument of a logarithm is or can written as a quotient (fraction or division), then this property let's you rewrite it as the difference of the logarithms of the numerator and denominator (divident and divisor).

They allow us to take a single logarithm and rewrite it as the sum or difference of logarithms. Let's see how they work on your problems.

1. log(4x)
There are many ways to express 4x as a product/multiplication. So there are many ways to rewrite log(4x). Here are a few:
log(4x) = log(4*x) = log(4) + log(x)
log(4x) = log(2*2x) = log(2) + log(2x)
log(4x) = log(16x*(1/4) = log(16x) + log(1/4)
We could also write 4x as a quotient and use the second property. For example:
log(4x) = log(20x/5) = log(20x) - log(5)

2. ln(15)
A product we could use for 15 could be 3*5 so
ln(15) = ln(3*5) = ln(3) + ln(5)
A quotient we could use for 15 could be 45/3 do
ln(15) = ln(45/3) = ln(45) - ln(3)

3. log(3/xy)
Again, there are many many ways we could do this. Probably the most obvious is to start with the second property:
log(3/xy) = log(3) - log(xy)
You might be able to stop here. But, since the argument of the second logarithm is a product, we can use the first property on it. (Note the use of parentheses. They are especially important when replacing one logarithm with an expression of two logarithms!)
log(3/xy) = log(3) - (log(x) + log(y))}}}
And because of the "-" in front of the parentheses, we should subtract both of the terms inside:
log(3/xy) = log(3) - log(x) - log(y)}}}
(Without the use of parentheses, it would be very easy to end up with a "+" in front of log(y)!)