Chapter 9.1, Problem 57E

To determine

To Explain: The logarithms of each term of a geometric sequence form an arithmetic sequence.

Explanation of Solution

Given Information: $\left\{a_n\right\}$ is a geometric sequence.

Concept Used:

 • The $n^{th}$ term of a geometric sequence with first term $a$ and common ratio $r$ is $a_n=a{r}^{n-1}$

 • $\log (p^q)=q\log (p)$

 • $\log (p)+\log (q)=\log (pq)$

 • $\log (p)-\log (q)=\log \left(\frac{p}{q}\right)$

The $n^{th}$ term and $n+1^{th}$ term of the geometric sequence are

  $a_n=a{r}^{n-1}$

  $a_{n+1}=a{r}^n$

Then

  $\log a_n=\log a{r}^{n-1}=\log a+\log r^{n-1}=\log a+(n-1)\log r$

  $\log a_{n+1}=\log a{r}^n=\log a+\log r^n=\log a+n\log r$

The difference between the logarithms of the two terms is therefore

  $\log a_{n+1}-\log a_n=\log a+n\log r-\log a-(n-1)\log r$

  $=n\log r-(n-1)\log r$

  $=\underset{\_}{\log r}$

It means the difference between the logarithms of any two consecutive terms is constant and is equal to $\log r$ .

Therefore, the logarithms of the terms of a geometric sequence form an arithmetic sequence with common difference $\log r$ .