Problem 88 - Algebra and Trigonometry

Question

Logarithms of a Geometric Sequence If \(a_{1}, a_{2},\) \(a_{3}, \ldots\) is a geometric sequence with a common ratio \(r>0\) and \(a_{1}>0,\) show that the sequence $$ \log a_{1}, \log a_{2}, \log a_{3}, \ldots $$ is an arithmetic sequence, and find the common difference.

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \(\log r\).

Step 1: Define the Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \(r\). The first term is \(a_1\). Hence, the terms of the sequence are \(a_1\), \(a_1r\), \(a_1r^2\), \(a_1r^3\), and so on.

Step 2: Express the Sequence with Logarithms

Take the logarithm of each term in the geometric sequence. The sequence becomes \(\log a_1\), \(\log (a_1 r)\), \(\log (a_1 r^2)\), \(\log (a_1 r^3)\), etc.

Step 3: Apply the Properties of Logarithms

Use the property of logarithms that \(\log(xy) = \log(x) + \log(y)\) to rewrite each term:\[\log(a_1 r^n) = \log a_1 + \log(r^n) = \log a_1 + n\log r\] for the \(n\)-th term.

Step 4: Simplify the Arithmetic Sequence

From Step 3, the sequence can be reduced to the form:\[\log a_1, \log a_1 + \log r, \log a_1 + 2\log r, \log a_1 + 3\log r, \ldots\]This is an arithmetic sequence with the first term \(\log a_1\) and a common difference of \(\log r\).

Step 5: Identify the Common Difference

The common difference in the logarithmic sequence is \(\log r\), as each term increases by this amount from the previous term.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is obtained by multiplying the previous number by a constant, called the common ratio. This ratio is denoted by \(r\) and must be non-zero. For example, if we start with a first term \(a_1\), then the second term is \(a_1r\), the third term is \(a_1r^2\), and so on. This consistent pattern makes geometric sequences easy to predict and calculate.

One of the unique features of geometric sequences is that the ratio between any two successive terms is the same throughout, which distinguishes them from arithmetic sequences where the difference between terms is constant, not their ratio. These sequences are important in many areas like finance for calculating interest and in science for understanding exponential growth.

Arithmetic Sequence

In an arithmetic sequence, each number in the list is obtained by adding a fixed number, known as the common difference, to the preceding term. If we start with a first term \(a_1\), the sequence proceeds as \(a_1, a_1 + d, a_1 + 2d, a_1 + 3d\), and so on, where \(d\) is the common difference.

The characteristic feature of an arithmetic sequence is that the difference between any two consecutive terms remains constant. This makes them quite straightforward to analyze and understand. Arithmetic sequences appear frequently in daily life and applications, such as when calculating the total distance covered by an object moving at a uniform speed or when determining the number of items in a pattern that increases by a constant amount.

Logarithm Properties

Logarithms are powerful mathematical tools that help in handling exponential relationships, and they have several properties that simplify calculations. One key property states that the logarithm of a product is the sum of the logarithms of the factors: \( \log(xy) = \log(x) + \log(y) \). This property allows us to linearize multiplicative relationships, turning them into additive ones, which are easier to manage.

Another important property is the power rule, \( \log(x^n) = n \log(x) \), which helps in simplifying the expression of powers by pulling the exponent down as a multiplier. These properties play a critical role in transforming a geometric sequence into an arithmetic sequence, as seen in the problem, by helping to express exponential terms in a more manageable, additive form.

• This is why logarithms are not just for solving equations, but are immensely useful in data analysis, especially when dealing with exponential growth or decay.