To solve the problem where [tex]\(\log x\)[/tex], [tex]\(\log y\)[/tex], and [tex]\(\log z\)[/tex] are consecutive terms in an arithmetic progression (AP), we follow these steps:
1. Understanding Arithmetic Progression (AP):
- In an AP, the difference between consecutive terms is constant. Let’s denote this common difference by [tex]\( d \)[/tex].
- Therefore, the terms can be expressed as:
[tex]\[
\log y - \log x = d \quad \text{and} \quad \log z - \log y = d.
\][/tex]
2. Setting Up Equations Based on AP Definition:
- From the definitions above, we can write:
[tex]\[
\log y - \log x = \log z - \log y.
\][/tex]
3. Simplifying the Equation:
- By simplifying the equation above:
[tex]\[
\log y - \log x = \log z - \log y,
\][/tex]
we can add [tex]\(\log y\)[/tex] to both sides:
[tex]\[
2 \log y = \log x + \log z.
\][/tex]
4. Applying Logarithm Properties:
- Using the properties of logarithms, specifically [tex]\(\log a + \log b = \log (ab)\)[/tex], we can rewrite the equation as:
[tex]\[
2 \log y = \log (x z).
\][/tex]
5. Exponentiating Both Sides:
- We can simplify this further by recognizing the properties of logarithms:
[tex]\[
\log (y^2) = \log (x z).
\][/tex]
Since the logarithms are equal, their arguments must also be equal:
[tex]\[
y^2 = x z.
\][/tex]
6. Conclusion - Geometric Progression:
- The relationship [tex]\( y^2 = x z \)[/tex] indicates that [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] are in a geometric sequence.
- In a geometric progression, the square of the middle term is equal to the product of the two outer terms, which is exactly what we have shown here.
Therefore, we have demonstrated that if [tex]\(\log x\)[/tex], [tex]\(\log y\)[/tex], and [tex]\(\log z\)[/tex] are consecutive terms in an arithmetic progression, then [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] must be in a geometric sequence.
