Problem 31 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \log _{\sqrt[5]{5}}^{2} \sqrt{5}-\log _{\sqrt[3]{5}}(5 \sqrt{5})+\log _{(\sqrt{3}+1)}(4+2 \sqrt{3}) . $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{2}\log_{\sqrt[5]{5}^2} 5 -\log_{\sqrt[3]{5}} 5^1 - \frac{1}{2} \log_{\sqrt[3]{5}} 5 + 2\).

Step 1: Identify properties of logarithm to apply on each term

In the given expression, we have three logarithm terms which should be simplified: - \(\log_{\sqrt[5]{5}^2} \sqrt{5}\) - \(-\log_{\sqrt[3]{5}}(5 \sqrt{5})\) - \(\log_{(\sqrt{3}+1)}(4+2 \sqrt{3})\) We need to identify and apply the logarithmic properties such as product rule, quotient rule, power rule, and change of base, if needed, to simplify each term.

Step 2: Simplify the first term

To simplify the first term \(\log_{\sqrt[5]{5}^2} \sqrt{5}\), we will use the power rule of logarithm: \(m \cdot \log_b a = \log_b a^m\) \( \log_{\sqrt[5]{5}^2} \sqrt{5} = \frac{1}{2}\log_{\sqrt[5]{5}^2} 5\) Here, we have used the power rule to take the power of 5 outside of the log in the form of its fraction.

Step 3: Simplify the second term

To simplify the second term \(-\log_{\sqrt[3]{5}}(5 \sqrt{5})\), we will first use the product rule of logarithm, and then the power rule: Product rule: \(\log_b (a \cdot c) = \log_b a + \log_b c\) Power rule: \(m \cdot \log_b a = \log_b a^m\) \(-\log_{\sqrt[3]{5}}(5 \sqrt{5}) = -\log_{\sqrt[3]{5}} 5 - \log_{\sqrt[3]{5}} \sqrt{5} = -\log_{\sqrt[3]{5}} 5^1 - \frac{1}{2} \log_{\sqrt[3]{5}} 5 \)

Step 4: Simplify the third term

To simplify the third term \(\log_{(\sqrt{3}+1)}(4+2 \sqrt{3})\), we don't need to use any properties of logarithm since both terms are in their simplest form. We do, however, need to recognise that the base and the argument are conjugates: \((\sqrt{3} + 1)(\sqrt{3} - 1) = 2\) Since the argument is twice the base, we know that: \( \log_{(\sqrt{3}+1)}(4+2\sqrt{3}) = 2\)

Step 5: Combine the simplified terms

Now that we have simplified each term, we can combine them: \(\frac{1}{2}\log_{\sqrt[5]{5}^2} 5 -\log_{\sqrt[3]{5}} 5^1 - \frac{1}{2} \log_{\sqrt[3]{5}} 5 + 2\) This expression cannot be simplified further as all three logarithmic terms have different bases, so this is the final answer.

Key Concepts

Logarithmic Properties

Logarithmic functions have several unique properties that help simplify complex expressions. These fundamental properties include:

• Product Rule: Used when multiplying two numbers. Transform the log of a product into the sum of logs: \[ \log_b(m \cdot n) = \log_b m + \log_b n \]
• Quotient Rule: Used for division. Divide the values and convert it into the difference of logs: \[ \log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n \]
• Power Rule: Helps to manage exponents by bringing them as a coefficient: \[ n \cdot \log_b m = \log_b m^n \]
• Change of Base Formula: Used to switch bases:\[ \log_b m = \frac{\log_k m}{\log_k b} \]

These properties are tools for simplifying logarithmic expressions and in the given exercise, both the product and power rules are prominently featured to streamline calculations.

Product Rule

The product rule for logarithms is an essential tool for simplifying expressions when the argument of the log is a product of two numbers. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In mathematical terms:\[ \log_b (m \cdot n) = \log_b m + \log_b n \]This allows us to break down more intricate logs into simpler parts.

In step 3 of the original exercise, this product rule was used to simplify \( -\log_{\sqrt[3]{5}} (5 \sqrt{5}) \) into individual components:

• \( -\log_{\sqrt[3]{5}} 5 \), and
• \( -\log_{\sqrt[3]{5}} \sqrt{5} \)

By applying the product rule, handling each term separately becomes more straightforward, especially when combined with other properties like the power rule.

Power Rule

The power rule of logarithms provides an easy way to manage logarithmic expressions with exponents. It states that the log of a number raised to an exponent can be represented as the exponent times the log of the base number:\[ n \cdot \log_b a = \log_b (a^n) \]This rule is beneficial when simplifying logs where an argument has a power.

In the original problem's second step, the power rule was pivotal for simplifying the term \( \log_{\sqrt[5]{5}^2} \sqrt{5} \). By expressing \( \sqrt{5} \) as \( 5^{1/2} \), and leveraged the power rule to move the exponent of \( \log_5 \) out front, the expression became :\[ \frac{1}{2} \log_{\sqrt[5]{5}^2} 5 \]This simplification helps avoid direct complex calculations, streamlining the process of simplification.

Change of Base Formula

The change of base formula is incredibly useful for evaluating or simplifying logs when the base isn't straightforward. This formula allows you to convert a logarithm from one base to another:\[ \log_b a = \frac{\log_k a}{\log_k b} \]Here, \(k\) can be any positive number, commonly 10 or \(e\), which makes the computation easier, especially on calculators or in algebraic manipulations.

The change of base formula wasn't directly used in the provided exercise, but it is essential to understand its applicability for simplifying logarithmic terms across different bases. This tool might be necessary if any further computation required uniform bases or for solving equations explicitly involving logarithms with unconventional bases.