Problem 72 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Prove that } \frac{\log _{a} n}{\log _{a b} n}=1+\log _{a} b $$

Step-by-Step Solution

Answer

The given statement \(\frac{\log_a n}{\log_{ab} n} = 1 + \log_a b\) has been proven to be true by applying logarithmic identities, simplification, and reciprocal properties of fractions.

Step 1 - Express denominator logarithm as a sum

Use the logarithmic identity \(\log_a (b*c) = \log_a b + \log_a c\) to transform \(\log_{ab} n\). Thus, \(\log_{ab} n = \log_a n / (\log_a a + \log_a b)\)

Step 2 - Simplify the denominator expression

Notice that from the properties of logarithms, \(\log_a a = 1\). Substitute this into the existing equation to get \(\log_{ab} n = \log_a n / (1 + \log_a b)\)

Step 3 - Swap numerator and denominator

Notice that the amount \(\log_a n / \log_{ab} n\) can be rewritten as the reciprocal of \(\log_{ab} n / \log_a n\), according to the property of fractions. Thus, we conclude that the statement under evaluation suffices to \(\frac{\log_a n}{\log_{ab} n} = (1 + \log_a b)\).

Key Concepts

Properties of Logarithms

The properties of logarithms are fundamental rules that help simplify complex logarithmic expressions. These rules aid in transforming and calculating logarithmic terms with ease. The main properties include:

• Product Rule: \( \log_a (b \cdot c) = \log_a b + \log_a c \)
• Quotient Rule: \( \log_a \left( \frac{b}{c} \right) = \log_a b - \log_a c \)
• Power Rule: \( \log_a (b^c) = c \cdot \log_a b \)
• Change of Base Formula: \( \log_a b = \frac{\log_c b}{\log_c a} \)

One vital property highlighted in this exercise is the identity used to express logarithms with different bases. For example, using the change of base formula, students can rewrite logarithm expressions in terms of another base, making complex calculations more manageable.
Understanding these properties is crucial as they serve as building blocks for solving advanced logarithmic equations. Recognizing when and how to apply these rules will simplify the given problems and assist in deriving solutions more systematically.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverse operations of exponentiation. They express how many times a base must be multiplied to reach a certain number.
In the given exercise, the expression \( \frac{\log_a n}{\log_{ab} n} \) involves logarithms with bases \( a \) and \( ab \). Tackling such expressions often requires applying the properties of logarithms to transform complex terms into simpler forms to facilitate easier computation and evaluation.
Given the task to prove the statement as in the original exercise, one learns how different bases in logarithmic expressions can be manipulated using known identities. By expressing \( \log_{ab} n \) in terms of \( \log_a n \) and \( \log_a b \), it demonstrates the flexibility of rewriting logarithmic terms, making the expressions less complex and more approachable for solving.
Successfully solving logarithmic expressions requires a strong grasp of logarithmic properties and an understanding of the relationships between the numbers involved.

Simplifying Logarithms

Simplifying logarithms is an essential skill for efficiently solving logarithmic equations. Through simplification, complex expressions are reduced to more straightforward forms using logarithmic identities and properties.
In the provided solution steps, we see how simplification is achieved by employing key logarithmic properties. By recognizing that \( \log_a a = 1 \), the complex expression \( \frac{\log_a n}{\log_a a + \log_a b} \) is reduced significantly.
The simplification process involves:

• Using identities such as \( \log_a a = 1 \) to substitute and reduce expression parts.
• Applying properties systematically to break down terms into sums or differences.
• Reworking expressions to more manageable fractions that inform the outcome.

Simplifying isn't just a step-by-step mechanical process. It requires logical reasoning and recognizing when it is necessary to apply the correct identity or property. Understanding how to simplify logarithms not only helps in solving the problem efficiently but also in proving results like the one given in the exercise.