Question

Assuming x and y and z are positive ,use properties of logarithm to write expression as a sum or difference of logarithms or multiples of logarithms. 13.log x + log y 14. log x + log y?

Answer 1

`log(x)+log(y) = log(xy)`

Explanation:

`log(x)+log(y) = log(x+y)` is a basic property of logarithms, along with

• `log(x)-log(y) = log(x/y)`
• `log(x^a) = alog(x)`
• `log_x(x) = 1`

To derive the property used in the given question, recall that `log_a(x)` is defined as the unique value fulfilling `a^(log_a(x)) = x`.

Then `log_a(xy)` is the unique value fulfilling `a^(log_a(xy)) = xy`.

Now, taking `a` to the power of `log_a(x)+log_a(y)`, we have

`a^(log_a(x)+log_a(y)) = a^(log_a(x))a^(log_a(y))=xy`

As `log_a(x)+log_a(y)` shares a property which is unique to `log_a(xy)`, it must be that `log_a(x)+log_a(y) = log_a(xy)`.