Understanding the Logarithm of Consecutive Integers

The problem asks us to simplify the expression log (1 + xz) given that x, y, and z are three consecutive positive integers. Let's break down the steps to find the solution.

Defining Consecutive Integers

Consecutive integers are numbers that follow each other in order. If we let the middle integer be y, then the integer before it is y-1 and the integer after it is y+1.

So, we can represent the three consecutive positive integers as:

• x = y - 1
• y = y
• z = y + 1

Since they are positive integers, y must be greater than 1 (y > 1) to ensure x (y-1) is also positive.

Simplifying the Expression 1 + xz

Now, we substitute the expressions for x and z into the term 1 + xz:

1 + xz = 1 + (y - 1)(y + 1)

We can use the difference of squares formula, (a - b)(a + b) = a^2 - b^2, where a = y and b = 1.

1 + xz = 1 + (y^2 - 1^2)

1 + xz = 1 + y^2 - 1

1 + xz = y^2

Applying the Logarithm

We need to find log (1 + xz). Using our simplified expression:

log (1 + xz) = log (y^2)

Using Logarithm Properties

One of the fundamental properties of logarithms states that log (a^b) = b log (a).

Applying this property to log (y^2):

log (y^2) = 2 log (y)

Conclusion

Therefore, if x, y, and z are three consecutive positive integers, log (1 + xz) simplifies to 2 log (y).

This matches the fourth option provided.