Problem 8 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(\log _{7} 2=x\), then \(\log _{49}(28)\) is (a) \(\left(x+\frac{1}{2}\right)\) (b) \(\left(x-\frac{1}{2}\right)\) (c) \(-\left(x-\frac{1}{2}\right)\) (d) \(-\left(x+\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

So, \(\log _{49}(28)\) can be expressed as \(4x+2\), which cannot be represented by any of the given choices. Therefore, none of the choices (a), (b), (c), or (d) are correct.

Step 1: Break Down the Expression into Simpler Logarithms

Using properties of logarithms, \(\log_{49}28\) can be written as \(2\log_{7}28\). Expressing it in terms of 2, the equation becomes \(2\log_{7}2 + 2\log_{7}14\).

Step 2: Substitute \(x\)

Substituting \(x=\log_{7}2\) in the equation results in \(2x+2\log_{7}14\).

Step 3: Simplify further using properties of logarithms

Simplify \(2x+2\log_{7}14\) into \(2x+2(\log_{7}2+ \log_{7}7)\). Notice that \(\log_{7}7\) equals 1.

Step 4: Substitute \(x\)

Substituting \(x= \log_{7}2\), the equation will be simplified into \(2x+ 2(x+1)\). After simplifying, it results in \(4x+2\).

Key Concepts

Logarithmic Equations

Understanding logarithmic equations is vital for solving a variety of mathematical problems. A logarithmic equation is one that involves a logarithm with a variable that needs to be solved for. Take, for instance, the equation \( \log_{b}(x) = y \), which can be rewritten using the definition of logarithms as \( b^y = x \).

To solve this type of equation, familiarity with exponent rules and the ability to isolate the variable are crucial. It's also common to utilize logarithmic properties to combine or decompose the logarithms in the equation. In the case of our exercise \( \log _{7} 2=x \) and \( \log _{49}(28) \), the solution involved breaking down the expression using the properties of logarithms, substituting variables, and simplifying.

Logarithmic Identities

Logarithmic identities, or properties, are rules that apply to logarithms which allow us to manipulate and simplify logarithmic expressions. Some fundamental identities include:

• The product rule: \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \)
• The quotient rule: \( \log_{b}(\frac{M}{N}) = \log_{b}(M) - \log_{b}(N) \)
• The power rule: \( \log_{b}(M^k) = k \log_{b}(M) \)

These identities are essential in transforming complex logarithmic expressions into simpler forms that are easier to work with. For example, in our exercise solution, the identity \( \log_{b}(M^k) = k \log_{b}(M) \) was used to simplify \( \log_{49}(28) \) into \( 2 \log_{7}(28) \) before further simplifying.

Change of Base Formula

The change of base formula is a crucial logarithmic property that enables you to rewrite a logarithm in terms of logs of another base. This is particularly handy when calculating logarithms without a calculator or converting to a base that simplifies the equation. The formula is given by:
\[ \log_{b}(M) = \frac{\log_{k}(M)}{\log_{k}(b)} \]
Using this formula, you can change the base of a logarithm to one that is more convenient for calculations. It's also instrumental in solving logarithmic equations where variables are present in the bases or the argument of the logarithms. While it was not directly used in the provided step-by-step solution, the change of base formula is an essential tool for understanding and solving various logarithmic equations.