Chapter 12.3, Problem 85E

PROVE: Logarithms of a Geometric Sequence If a₁, a₂,

a₃, … is a geometric sequence with a common ratio r > 0 and a₁ > 0, show that the sequence

$\log a_1,\log a_2,\log a_3,\mathrm{...}$

is an arithmetic sequence, and find the common difference.

To determine

To evaluate: The sequence $\log a_1,\log a_2,\log a_3,\cdots$ is a arithmetic sequence if $a_1,a_2,a_3,\cdots$ is a geometric sequence with common ratio r and find the common difference of the sequence $\log a_1,\log a_2,\log a_3,\cdots$.

Explanation of Solution

Proof:

The geometric sequence is given,

$a_1,a_2,a_3,\cdots$ (1)

The common ratio of this sequence is r.

The common ratio determines the ratio of second and first term of the sequence (1) is equal to the ratio of third term and second term of the sequence (1),

$r=\frac{a_2}{a_1}$

And,

$r=\frac{a_3}{a_2}$

So, the relation between terms is,

$\begin{array}{l}\frac{a_2}{a_1}=\frac{a_3}{a_2}\\ a_2{}^2=a_3\cdot a_1\end{array}$

The above relation defines the property of geometric series.

The sequence is given,

$\log a_1,\log a_2,\log a_3,\cdots$ (2),

If the above sequence is arithmetic sequence then it satisfies the relation $a_2{}^2=a_3\cdot a_1$ of sequence (1).

If the sequence is arithmetic then the subtraction of second and first term of the sequence (2) is equal to the subtraction of third term and second term of the sequence (2),

So, above statement gives below relation,

$\begin{array}{c}\log a_2-\log a_1=\log a_3-\log a_2\\ \log a_2+\log a_2=\log a_3+\log a_1\\ \log a_2{}^2=\log a_1\cdot a_3\\ a_2{}^2=a_1.a_3\end{array}$

The above relation satisfies the property of geometric sequence $a_1,a_2,a_3,\cdots$.

Thus, the sequence $\log a_1,\log a_2,\log a_3,\cdots$ is a arithmetic sequence.

The sequence $\log a_1,\log a_2,\log a_3,\cdots$ is an arithmetic sequence.

The common difference of this sequence is the difference of second term and first term,

$\begin{array}{c}d=\log a_2-\log a_1\\ =\log \left(\frac{a_2}{a_1}\right)\end{array}$

Thus the common difference of sequence is $\log \left(\frac{a_2}{a_1}\right)$.