Assume that x, y, and a are positive numbers. Use the properties of logarithms to write the expression log a 9 xy in terms of the logarithms of x and y.

a. 9logbx+9logby
b. 19 l o g b ( x + y )
c. 19 l o g b x + l o g b y
d. logbx+logby
e. 19 l o g b x + 19 l o g b y

Where does the b come from in all your answers?

Where does the 19 come from in all the answers?

I suspect a typo

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

regardless of the base, and considering the chance of typos, I'd go with (e)

To write the expression log a 9xy in terms of the logarithms of x and y, you need to use the properties of logarithms.

One property states that log a (mn) = log a m + log a n.

Another property states that log a (m^n) = n(log a m).

Using these properties, let's rewrite log a 9xy:

log a 9xy = log a (9 * x * y)

Now, using the first property, we can break down the multiplication:

log a (9 * x * y) = log a 9 + log a x + log a y

Therefore, the expression log a 9xy in terms of the logarithms of x and y is:

log a 9xy = log a 9 + log a x + log a y

So, the correct answer is option a. 9logbx+9logby.