If x, y, z are three consecutive positive integers, then log (1 + xz) is:

Step-by-Step Solution

To simplify the expression log(1 + xz), we first express x and z in terms of y, given that x, y, and z are consecutive positive integers.

1. Express consecutive integers in terms of y:

   • Let the three consecutive integers be: x = y - 1 y = y z = y + 1

2. Simplify the expression inside the logarithm (1 + xz):

   • Substitute x = y - 1 and z = y + 1 into the expression 1 + xz: 1 + (y - 1)(y + 1)

   • Using the difference of squares formula, (y - 1)(y + 1) = y^2 - 1.

   • So, 1 + xz = 1 + (y^2 - 1)

   • 1 + xz = y^2

3. Apply the logarithm:

   • The original expression is log(1 + xz).

   • Substituting our simplified result: log(y^2)

   • Using the logarithmic power rule, log(a^b) = b log(a), we can move the exponent 2 in front: 2 log(y)