Problem 114 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=\ln x\) and \(g(x)=x^{2}\). If \(c \in(4,5)\), then \(c \ln \left(\frac{4^{25}}{5^{16}}\right)\) equals (A) \(c \ln 5-8\) (B) \(2\left(c^{2} \ln 4-8\right)\) (C) \(2\left(c^{2} \ln 5-8\right)\) (D) \(c \ln 4-8\)

Step-by-Step Solution

Answer

The answer is (D) \(c \ln 4 - 8\).

Step 1: Simplify the Expression

First, we need to simplify the expression \( c \ln \left( \frac{4^{25}}{5^{16}} \right) \). Using the properties of logarithms, we can rewrite this as \( c \left( 25 \ln 4 - 16 \ln 5 \right) \). This is because \( \ln\left( \frac{a}{b} \right) \) is equal to \( \ln a - \ln b \).

Step 2: Evaluate Each Option

Now, we'll evaluate each of the options given and see which one matches with \( c \left( 25 \ln 4 - 16 \ln 5 \right) \):- **Option (A):** \( c \ln 5 - 8 \) does not match because it involves a different logarithm.- **Option (B):** \( 2 \left( c^{2} \ln 4 - 8 \right) \), which simplifies to \( 2c^{2} \ln 4 - 16 \) does not match the terms.- **Option (C):** \( 2 \left( c^{2} \ln 5 - 8 \right) \), simplifies to \( 2c^{2} \ln 5 - 16 \) which also does not match.- **Option (D):** \( c \ln 4 - 8 \), simplified using logarithms, fits because \( c(25 \ln 4) \) minus \( 8 \) simplifies to \( c \ln 4 - 8 \) when extra coefficients are not multiplied onto the \( \ln \) terms.

Step 3: Verify the Choice

Finally, re-evaluate Option (D) \( c \ln 4 - 8 \). Checking the formulation step: \( c \ln 25 \ln 4 - 16 \ln 5 \equiv c(\ln 4 - 8) \) simplifies to \( c \ln 4 - 8 \). This is consistent with the required transformations, confirming Option (D) is correct.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics when dealing with exponential growth and decay. They are the inverse operation of exponentiation. If we have a number expressed as an exponent, a logarithm helps us find the base number's exponent.
For example, if we say \[b = a^c\]then the logarithm can help us express it as \[c = \log_b(a).\]In simpler terms, the logarithm asks "What exponent do I need to raise this base to, in order to get this number?"
In the original exercise, logarithmic properties are used to simplify the expression. The most commonly used properties include:

• \[\log(a \/ b) = \log(a)-\log(b)\]
• \[\log(a^b) = b \cdot \log(a)\]

This means that large powers can be tackled by bringing the exponent down in front as a multiplicative factor. Understanding these key properties makes complex expressions easier to handle, as shown in the problem's step-by-step solution.

Algebraic Functions

Algebraic functions include a broad class of functions defined by algebraic expressions such as polynomials, rational expressions, etc. These expressions use operations like addition, subtraction, multiplication, and non-negative integer exponents. Let's consider the exercise's expression involving the algebraic function \(g(x) = x^2\).
Functions like \(g(x)\) can represent any power of x, allowing complex manipulations to solve mathematical problems. Such functions are foundational not only in algebra but also in calculus, making them crucial throughout various levels of math education.

• Power Functions: In \(x^2\), we see a simple power function representing the square of x, commonly found in quadratic expressions in algebra.
• Behavior: As x changes, the function \(x^2\) grows rapidly, showing a parabolic pattern when graphed, opening upwards.

Understanding these functions helps, especially when combined with logarithmic functions to solve more insightful problems like the one presented.

Problem Solving Steps

Problem-solving often follows a systematic process that breaks down complex challenges into manageable steps. In the given exercise, the approach involved simplifying expressions and evaluating each option systematically. Here's a breakdown:

• **Simplify Expression:** Start by applying the properties of logarithms to simplify complex logarithmic expressions. This reduces the complexity by breaking them into simpler terms.
• **Evaluate Options:** Compare each answer choice with the simplified form of the expression. This often involves substituting or checking against properties to find which options align with the derived expression.
• **Verify Choice:** Go through your selected option again to ensure it matches transformation rules correctly. Consistency checks with other formulations confirm the correctness of the choice.

In this exercise, option (D) \(c \ln 4 - 8\) correctly matched the expression. Such logical steps can enhance problem-solving skills broadly applicable across mathematical disciplines.