Problem 25 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the value of \(81^{\frac{1}{\log _{5} 3}}+27^{\log _{9} 36}+3^{\frac{4}{\log _{7} 9}}\)

Step-by-Step Solution

Answer

The short answer is \(Sum = 5^4 + 3^{3*2/(1+log _{3} 2)} + 3^{14}\). Further simplification might be more complex due to presence of logarithmic terms.

Step 1: Simplify the first term

The first term is \( 81^{\frac{1}{\log _{5} 3}} \). Here, observe that 81 is \(3^4\). So we can write this term as \( (3^4)^{\frac{1}{\log _{5} 3}} = 3^{4/ \log _{5} 3} \). From the properties of logarithms, \( \log _{5} 3 = \frac{1}{ \log _{3} 5} \). Replacing this, the first term simplifies to \( 3^{4 * \log _{3} 5} = 5^4 \).

Step 2: Simplify the second term

The second term is \( 27^{\log _{9} 36} \). Here, observe that 27 is \(3^3\) and 9 is \(3^2\). So we can write this term as \( (3^3)^ {\log_{(3^2)}{ (6^2)}} = 3^{3 * \log_{(3^2)}{ (6^2)}} = 3^{3*2\log _{3}6} = 3^{3*2*\frac{1}{ \log _{3} 6}} \). Since \(6= 2*3\), we can write \( \log _{3} 6 = \log _{3} (2*3)= 1 + \log_{3} 2\). Now replace this into the previous expression: \( 3^{3*2*(1/{1+log _{3} 2})}\), hence this term becomes \( 3^{3*2/(1+log _{3} 2)}\).

Step 3: Simplify the third term

Similar as before, here 3 is \(3^1\), so this third term can be rewritten as \( (3^1)^{\frac{4}{\log _{7} 9}} = 3^{1 *{\frac{4}{\log _{7} 9}}}= 3^{\frac{4}{\log _{7} 3^2}} =3^{\frac{4}{2\log _{7} 3}}=3^{2/{\log _{7} 3}} =3^{2*7}= 3^{14}\).

Step 4: Add all the terms

Finally, adding all terms up: \(Sum = 5^4 + 3^{3*2/(1+log _{3} 2)} + 3^{14}\)

Key Concepts

Exponential Functions

Exponential functions are essential mathematical tools in many areas of study. These functions involve constant exponents and a variable base. An exponential function can be expressed as \( f(x) = a^x \), where \(a\) is a positive real number, and \(x\) is any real number. In this exercise, we dealt with terms like \(81^{1/\log _{5} 3}\) and \(27^{\log _{9} 36}\) which are essentially exponential expressions.
The base can be rewritten using a common base to ease calculations.

• For example, 81 can be rewritten as \(3^4\), and 27 as \(3^3\). This conversion allows us to simplify terms by seeing them in terms of the same base.
• Once you express everything with the common base, it becomes easier to apply logarithmic properties to solve the expression.

Rewriting the bases helps in simplifying the problem by expressing it in a comparable form, showcasing the power of exponentiation.

Properties of Logarithms

Understanding logarithms is crucial when simplifying expressions like those in the original exercise. Logarithms are the inverse of exponential functions, where \( \log_b{a} \) asks the question, "To what power should \(b\) be raised to yield \(a\)?"
Some of the important properties include:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^k) = k \cdot \log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \) for any positive number \(k\).

In this exercise, the logarithms help by simplifying complex exponents. For instance, if you encounter a term like \( \log_5 3 \), using the change of base formula simplifies it to \( \frac{1}{\log_3 5} \). Strategic application of these properties makes solving logarithmic and exponential expressions much more straightforward.

Simplification Techniques

Simplification techniques are pivotal in reducing complex mathematical expressions into simpler forms that are easier to handle. Many times, simplifications involve combining properties of exponents and logarithms.
In the given exercise, these techniques include:

• **Changing Terms to Common Bases:** Converts expressions into simpler, comparable forms using a common base, like expressing 81 as \(3^4\) and 27 as \(3^3\).
• **Utilizing Logarithmic Properties:** Employs properties such as the power and change of base to transform and simplify expressions.
• **Adding Simplified Terms:** In the final step, after simplifying each exponential term separately, they are combined into a single, straightforward expression that is easier to compute or further analyze.

The goal of simplification is to understand and solve complex expressions by breaking them down into manageable parts and using mathematical properties effectively. These techniques not only help with calculating the solutions but also provide insight into the underlying structure of the expressions.