Problem 43 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(x^{18}=y^{21}=z^{28}=k\), then prove that, \(3,3 \log _{y} x, 3 \log _{z} y, 7 \log _{x} z\) are in A.P.

Step-by-Step Solution

Answer

After converting the given equalities to logarithmic form, we can rewrite each of the given terms by substituting the logs with the bases and arguments. They form an arithmetic progression, as proved.

Step 1 - Rewrite the given equations

Given \(x^{18} = y^{21} = z^{28} = k\), we can rewrite these equations as logarithmic expressions: \(\log_x{k} = 18\), \(\log_y{k} = 21\), and \(\log_z{k} = 28\)

Step 2 - Express the terms using a common base

Using properties of logarithms, we can express x, y, and z in terms of k: \(x = k^{1/18}\), \(y = k^{1/21}\), and \(z = k^{1/28}\). We can substitute these into each term: \(\log _{y} x = \log _{k^{1/21}} k^{1/18} = \frac{1}{18} / \frac{1}{21} = 21/18\), \(\log _{z} y = \log _{k^{1/28}} k^{1/21} = \frac{1}{21} / \frac{1}{28} = 28/21\), \(\log _{x} z = \log _{k^{1/18}} k^{1/28} = \frac{1}{28} / \frac{1}{18} = 18/28\)

Step 3 - Put the terms in A.P.

Finally, the exact values from step 2 need to be put in their respective places in the terms provided in the problem: The terms \(3, 3 * 21/18, 3 * 28/21, 7 * 18/28\), which respectively correspond to \(3, 3.5, 4, 4.5\), form an arithmetic progression.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics that help us work with numbers on a multiplicative scale.
They are the inverse operations of exponents, similar to how division is the inverse of multiplication.
In the context of the exercise, we dealt with expressions that could be rewritten in logarithmic form. For example, if we know that \(x^{18} = k\), we can express that as \(\log_x{k}=18\).
This indicates that raising \(x\) to the power of 18 results in \(k\). If you take the logarithm of both sides with base \(x\), you can isolate the exponent as \(18\).
**Properties of Logarithms**:

• The product rule: \(\log_b (mn) = \log_b m + \log_b n\)
• The quotient rule: \(\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n\)
• The power rule: \(\log_b (m^n) = n \cdot \log_b m\)

These are some of the vital properties that helped us transform and simplify the logarithmic expressions in solving this exercise.

Exponents

Exponents describe the number of times a number is multiplied by itself. In the equation \(x^{18} = y^{21} = z^{28} = k\), each base (\(x, y, \text{and} z\)) is raised to a certain power, creating equal expressions.
This intriguing equation shows how different exponential forms can represent the same value. Exponential expressions like \(x^{18}\) allow us to compare and solve for the bases \(x, y, \text{and} z\) through their exponents.
**Key Points About Exponents**:

• Exponent multiplication: \(a^m \cdot a^n = a^{m+n}\)
• Exponent division: \(a^m / a^n = a^{m-n}\)
• Power raising: \((a^m)^n = a^{m \cdot n}\)

Understanding these rules is crucial for manipulating and transforming expressions to find equivalent forms, much like how we expressed \(x, y, \text{and} z\) using a common base \(k\) in the exercise.

Sequences and Series

In mathematics, a sequence is an ordered list of numbers following a particular pattern. An arithmetic progression (AP) is a type of sequence where the difference between consecutive terms is constant.
In the given exercise, the progression of numbers derived from logarithmic transformations forms an AP. That is, the terms are structured such that the difference between any two adjacent terms is uniform.**Arithmetic Progression (AP) Characteristics**:

• The general form of an AP is \(a, a+d, a+2d, \ldots\), where \(a\) is the first term and \(d\) is the common difference.
• If three numbers are in AP, the middle one is always the average of the other two: \(b = \frac{a+c}{2}\).

In the exercise, we demonstrated how the logarithmic values transformed into \(3, 3.5, 4, 4.5\), with their sequence following the arithmetic pattern having a constant difference between terms, showcasing their membership in an arithmetic progression.