Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\frac{1}{\log _{a} x}+\frac{1}{\log _{c} x}=\frac{2}{\log _{b} x}\), then \(a, b, c\) are in (a) A.P. (b) G.P. (c) H.P. (d) A.G.P.

Step-by-Step Solution

Answer

Therefore, the correct answer is (b) G.P.

Step 1: Analyze the given equation

Notice that if we can use the property \(\frac{1}{\log_n m} = \log_m n\), the equation can be rewritten as \(\log_x a + \log_x c = 2 \log_x b\).

Step 2: Rewrite using logarithm properties

The sum of logs becomes the log of the product. Therefore, the equation becomes: \(\log_x (ac) = \log_x (b^2)\). This implies \(ac = b^2\).

Step 3: Evaluate the possible sequences

Comparing our result to the possible sequences, the relationship \(ac = b^2\) indicates a geometric progression. Its characteristic is that each term is a product of the previous term and a constant. Here, \(b\) is equivalent to the square root of the product of \(a\) and \(c\). In other words, \(b\) is the geometric mean of \(a\) and \(c\), confirming that the sequence is geometric.

Key Concepts

Logarithm Properties

Logarithms are a mathematical tool that helps us simplify complex multiplication and division operations. One of the essential properties to understand when working with logarithmic equations is how they can transform into other logarithmic forms. For example, the property \(\frac{1}{\log_n m} = \log_m n\) is incredibly helpful.

• This property allows us to switch the base and the argument of the logarithm, effectively flipping them.
• Understanding this can make solving logarithmic equations much simpler, as equations often require you to get common bases or arguments.

Another crucial property of logarithms is that the sum of logs is equal to the log of their product. This can be expressed as \(\log_x a + \log_x b = \log_x (a \times b)\). Breaking down these complex logarithmic expressions into simpler products and sums can often reveal patterns in the numbers, such as identifying progression types or sequences.

Geometric Progression (G.P.)

Geometric Progression is all about the consistent ratio between consecutive terms in a sequence. If each term of a sequence can be found by multiplying the previous term by a constant, then you have a G.P. The general form of a G.P. is: \[a, ar, ar^2, ..., ar^n, ...\]

• Here, \(a\) is the first term, and \(r\) is the common ratio.
• Each subsequent term is created by multiplying the previous term by \(r\).

In our exercise, the expression \(ac = b^2\) suggests that \(b\) is the geometric mean of \(a\) and \(c\). This confirms that \(a, b, c\) are in G.P. since \(b = \sqrt{ac}\), which falls in line perfectly with the definition of a G.P. where the middle term is the geometric mean between the other two terms.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific rule or pattern. Understanding sequences is vital in solving many mathematical problems and helps in recognizing patterns within data or formulas.Sequences can come in different forms:

• Arithmetic Sequence (A.P.): Where each term increases by a common difference.
• Geometric Sequence (G.P.): Where each term is multiplied by a constant ratio, as described earlier.
• Harmonic Sequence (H.P.): Obtained by taking the reciprocals of an arithmetic sequence.

Recognizing these patterns allows us to determine the type of sequence, predict future terms, and identify relationships between terms. In the context of the homework exercise, identifying the relationship \(ac = b^2\) helped to confirm that \(a, b, c\) are in geometric progression, illustrating the value of understanding these sequence types well.