Assume that x, y, and b are positive numbers. Use the properties of logarithms to write the expression logb ^8xy in terms of the logarithms of x and y.
a. logb^8 + logb x + logb^y
b. logb^8+logbx
c. logb^8+logby
d. logb^8 + log8 x + log8^y
e. logb^8+log8xy
To write the expression logb^8xy in terms of the logarithms of x and y, we can use the properties of logarithms to break it down into separate logarithms.
The properties of logarithms that we will use are:
1. logb(xy) = logb(x) + logb(y) - the product rule of logarithms
2. logb(x^m) = m * logb(x) - the power rule of logarithms
Now let's simplify the expression logb^8xy:
logb^8xy = logb(8xy)
Using the product rule of logarithms, we can split 8xy into two separate logarithms:
logb(8) + logb(x) + logb(y)
Applying the power rule of logarithms to logb(8), we can simplify it further:
3 * logb(2) + logb(x) + logb(y)
So, the expression logb^8xy in terms of the logarithms of x and y is:
c. logb^8 + logby