Problem 79 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a, b, c\) (all +ive) are the \(p\) th, \(q\) th, \(r\) th, terms respectively of a geometric progression, then prove that \(\left|\begin{array}{lll}\log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1\end{array}\right|=0\)

Step-by-Step Solution

Answer

The determinant of the given matrix is zero, hence proved.

Step 1: Understand the problem

A Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. So, the \(p\) th, \(q\) th, \(r\) th terms will be \(ar^{p-1}\), \(ar^{q-1}\), \(ar^{r-1}\) respectively, where \(a\) is the first term and \(r\) is the common ratio.

Step 2: Apply the properties of logarithms

Using the properties of logarithms, we can write that \(\log(ar^{p-1}) = \log a + (p-1)\log r\). Similar expressions can be written for \(\log b\) and \(\log c\).

Step 3: Substitute into determinant

Now, substitute these expressions into the determinant. The determinant will now become $$\left|\begin{array}{lll} \log a + (p-1)\log r & p & 1 \\ \log a + (q-1)\log r & q & 1 \\ \log a + (r-1)\log r & r & 1 \end{array}\right|=0$$. This can be simplified by subtracting \((p-1)\log r\) from the first row, \((q-1)\log r\) from the second row and \((r-1)\log r\) from the third row.

Step 4: Simplify the determinant

On simplifying the determinant, we get $$\left|\begin{array}{lll} \log a & p & 1 \\ \log a & q & 1 \\ \log a & r & 1 \end{array}\right|=0$$. Now this is a determinant of 3 rows with identical first column, hence, the value of this determinant is zero, which proves the given statement.

Key Concepts

Properties of Logarithms

Logarithms are an indispensable tool in mathematics, especially when dealing with growth patterns, such as those found in geometric progressions. The properties of logarithms allow us to manipulate expressions and solve equations that would otherwise be complex.

At its core, a logarithm answers the question: to what power must we raise a given base to obtain a certain number? For example, the expression \(\text{log}_b(x)\) means what power should base \(b\) be raised to in order to get \(x\). Here are some essential properties of logarithms that are widely used in mathematics:

• Product Property: For any positive numbers \(a\), \(b\), and base \(c\), the logarithm of a product is the sum of the logarithms: \(\text{log}_c(a\text{ \times } b) = \text{log}_c(a) + \text{log}_c(b)\).
• Quotient Property: For any positive numbers \(a\), \(b\), and base \(c\), the logarithm of a quotient is the difference of the logarithms: \(\text{log}_c(\frac{a}{b}) = \text{log}_c(a) - \text{log}_c(b)\).
• Power Property: For any positive number \(a\), exponent \(p\), and base \(c\), the logarithm of a power is the exponent times the logarithm of the base: \(\text{log}_c(a^p) = p\text{ \times }\text{log}_c(a)\).

These properties simplify complex logarithmic expressions and allow us to transform multiplicative relationships into additive ones, making calculus, among other areas of mathematics, more manageable. As seen in our original problem regarding geometric progression, the properties of logarithms were key to converting the terms into a form that could be substituted into a determinant for proof.

Solving Determinants

In linear algebra, determinants play a pivotal role in understanding the properties of matrices. A determinant is a scalar value that is a function of a square matrix and can provide essential information about the matrix, such as whether the matrix is invertible or not.

When solving determinants, a variety of methods can be employed, including the method of co-factors, Gaussian elimination, and expansion by minors. The determinant of a 3x3 matrix, such as the one in our geometric progression exercise, is often calculated using the rule of Sarrus or a similar strategy. The value of a determinant provides insight into the nature of the linear system it represents; for instance, when a determinant equals zero, it indicates that the system has no unique solution.

In the case of our exercise, the determinant's simplification to a form in which all elements of a column are identical immediately tells us that it has a value of zero. This occurrence showcases a critical property: if any row or column of a matrix is a scalar multiple of another, the determinant of that matrix is zero.

Understanding how to find and interpret the determinant is crucial for fields such as physics, engineering, and economics, where systems of equations and matrix transformations are foundational.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a mathematical statement. Proofs are the backbone of mathematics, solidifying conjectures by building on axioms, definitions, and previously established theorems.

The process involves deducing conclusions by applying logical reasoning to given premises. There are several proof techniques, including direct proof, indirect proof, proof by contradiction, and proof by induction, each suited to different kinds of problems.

In our discussion of geometric progressions, the proof begins by leveraging the properties of logarithms and the calculation of a determinant. The proof concludes by showing that the determinant equates to zero, which is consistent with the characteristics of the matrix created from the logarithms of the terms in the geometric progression. This simple yet elegant argument illustrates the satisfaction and clarity that comes from a well-crafted proof, which is both beneficial for advancing mathematical understanding and crucial for demonstrating the validity of a concept within mathematics.