Arithmetic Progression And Logarithms A Comprehensive Guide

This article delves into the fascinating interplay between arithmetic progressions and logarithms, providing a detailed solution to a multi-part problem. We will explore how to determine the number of terms in an arithmetic progression given its sum and how to manipulate logarithmic expressions without the aid of a calculator. This exploration is designed to enhance your understanding of these fundamental mathematical concepts.

Part (a): Finding the Number of Terms in an Arithmetic Progression

In this section, we address the first part of the problem, which involves finding the number of terms in an arithmetic progression. Arithmetic progressions are sequences of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference. The problem states that the first three terms of an arithmetic progression are given as lg3, 3lg3, and 5lg3. Our goal is to determine the value of 'n', which represents the number of terms in the progression, given that the sum of these 'n' terms can be expressed as 256 lg 81.

To begin, let's identify the key components of the arithmetic progression. The first term, often denoted as 'a', is lg3. The common difference, denoted as 'd', can be found by subtracting the first term from the second term (or the second term from the third term). In this case, d = 3lg3 - lg3 = 2lg3. Now that we have the first term and the common difference, we can utilize the formula for the sum of an arithmetic progression to solve for 'n'. The sum of the first 'n' terms of an arithmetic progression, denoted as Sn, is given by the formula: Sn = (n/2) * [2a + (n-1)d]. We are given that Sn = 256 lg 81. Substituting the values of 'a' and 'd' into the formula, we get:

256 lg 81 = (n/2) * [2(lg3) + (n-1)(2lg3)]

Our next step involves simplifying this equation and solving for 'n'. We can begin by simplifying the expression inside the brackets:

256 lg 81 = (n/2) * [2lg3 + 2nlg3 - 2lg3]

Notice that the terms 2lg3 and -2lg3 cancel each other out, simplifying the equation further:

256 lg 81 = (n/2) * [2nlg3]

256 lg 81 = n^2 lg3

Now, we need to address the logarithmic terms. We know that 81 can be expressed as 3^4. Therefore, lg 81 = lg (3^4). Using the logarithmic power rule, which states that lg(a^b) = b * lg(a), we can rewrite lg 81 as 4lg3. Substituting this into our equation, we get:

256 * 4lg3 = n^2 lg3

1024lg3 = n^2 lg3

We can now divide both sides of the equation by lg3, which eliminates the logarithmic term and simplifies the equation to:

1024 = n^2

To solve for 'n', we take the square root of both sides:

n = √1024

n = 32

Therefore, the value of 'n', which represents the number of terms in the arithmetic progression, is 32. This completes the solution for part (a) of the problem. We have successfully utilized the properties of arithmetic progressions and logarithms to determine the number of terms given the sum and the initial terms of the progression. The key steps involved identifying the first term and common difference, applying the formula for the sum of an arithmetic progression, simplifying the equation using logarithmic properties, and solving for 'n'.

Part (b): Logarithmic Expression Evaluation Without a Calculator

In the second part of this question, we are presented with a logarithmic expression that we must evaluate without the use of a calculator. This exercise is designed to test our understanding of logarithmic properties and our ability to manipulate logarithmic expressions. The specific expression to be evaluated is not provided in the initial prompt, so let's assume for the sake of demonstration, we are asked to simplify and evaluate the expression:

log₂(8√2)

This example will allow us to illustrate the techniques involved in simplifying logarithmic expressions without relying on a calculator. The core principle in evaluating logarithmic expressions without a calculator is to express the arguments of the logarithms in terms of the base. In this case, the base of the logarithm is 2, so we want to express 8 and √2 as powers of 2.

Let's begin by breaking down the expression step by step. We know that 8 can be expressed as 2³, and √2 can be expressed as 2^(1/2). Substituting these into the expression, we get:

log₂(8√2) = log₂(2³ * 2^(1/2))

Now, we can use the property of exponents that states a^m * a^n = a^(m+n). Applying this property to the expression inside the logarithm, we have:

log₂(2³ * 2^(1/2)) = log₂(2^(3 + 1/2))

To add the exponents, we need a common denominator. So, we rewrite 3 as 6/2:

log₂(2^(6/2 + 1/2)) = log₂(2^(7/2))

Now, we can use another fundamental logarithmic property, which states that logₐ(a^b) = b. This property is crucial for simplifying logarithmic expressions where the base of the logarithm matches the base of the argument. Applying this property to our expression, we get:

log₂(2^(7/2)) = 7/2

Therefore, the simplified value of the expression log₂(8√2) is 7/2 or 3.5. This demonstrates how we can evaluate logarithmic expressions without a calculator by expressing the arguments in terms of the base and utilizing logarithmic properties.

Let's consider another example to further illustrate this technique. Suppose we are asked to evaluate:

log₃(9/√3)

Again, we aim to express 9 and √3 as powers of 3. We know that 9 = 3² and √3 = 3^(1/2). Substituting these into the expression, we get:

log₃(9/√3) = log₃(3² / 3^(1/2))

Now, we use the property of exponents that states a^m / a^n = a^(m-n). Applying this property, we have:

log₃(3² / 3^(1/2)) = log₃(3^(2 - 1/2))

To subtract the exponents, we rewrite 2 as 4/2:

log₃(3^(4/2 - 1/2)) = log₃(3^(3/2))

Finally, we apply the logarithmic property logₐ(a^b) = b:

log₃(3^(3/2)) = 3/2

So, the simplified value of log₃(9/√3) is 3/2 or 1.5. These examples highlight the importance of understanding logarithmic properties and being able to manipulate exponents to evaluate logarithmic expressions without a calculator. The key is to express the arguments in terms of the base and then apply the appropriate logarithmic properties to simplify the expression.

In summary, evaluating logarithmic expressions without a calculator requires a solid understanding of logarithmic properties and the ability to manipulate exponents. By expressing the arguments in terms of the base and applying properties such as logₐ(a^b) = b, logₐ(mn) = logₐ(m) + logₐ(n), and logₐ(m/n) = logₐ(m) - logₐ(n), we can simplify complex logarithmic expressions and arrive at the correct answer. This skill is crucial for developing a deeper understanding of logarithms and their applications in mathematics and other fields.

Conclusion

Throughout this article, we have explored the interplay between arithmetic progressions and logarithms, successfully solving a multi-part problem. We determined the number of terms in an arithmetic progression given its sum and the first few terms. Furthermore, we demonstrated how to evaluate logarithmic expressions without the aid of a calculator, emphasizing the importance of understanding and applying logarithmic properties. These exercises underscore the fundamental nature of arithmetic progressions and logarithms in mathematics and highlight the importance of mastering these concepts for further studies in mathematics and related fields.