Problem 18 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \log _{9} 27-\log _{27} 9 $$

Step-by-Step Solution

Answer

The simplified expression for the given logarithmic expression is \(\frac{5}{6}\).

Step 1: Rewrite logarithms in terms of base 3

We have two logarithms with bases 9 and 27, which can be expressed as powers of 3. Therefore, we can rewrite the logarithms in terms of base 3: $$ \log_9{27} - \log_{27}9 = \frac{\log_3{27}}{\log_3{9}} - \frac{\log_3{9}}{\log_3{27}} $$

Step 2: Find the values of the new logarithms

Next, we'll find the value of the logarithms with base 3: $$ \log_3{27} = 3 \quad \text{since} \quad 3^3 = 27, \\ \log_3{9} = 2 \quad \text{since} \quad 3^2 = 9. $$

Step 3: Substitute the values of the logarithms

Now, we can substitute the values of the logarithms back into the expression: $$ \frac{3}{2} - \frac{2}{3} $$

Step 4: Find the common denominator and simplify

Finally, we will find the common denominator (6 in this case) and simplify the expression: $$ \frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} $$ $$ \frac{9-4}{6} = \frac{5}{6} $$ So the simplified expression is \(\frac{5}{6}\).

Key Concepts

Logarithmic Identities

Logarithmic identities are fundamental tools in the simplification and calculation of expressions involving logarithms. One of the key identities deals with converting complex logarithmic expressions into simpler ones with a common base.

• The change of base formula allows us to rewrite a logarithm in terms of a new base: \[\log_{b}{a} = \frac{\log_{k}{a}}{\log_{k}{b}}\]where \( k \) is the new base of the logarithms. This identity is particularly useful when needing to perform arithmetic operations on logarithms with different bases, as in the given exercise.
• By utilizing this identity, we can transform the expression \(\log_9{27} - \log_{27}9\) into a single base expression.
• This simplifies the comparison and subtraction of the two terms by eliminating the need to work directly with various bases.

Having a solid grasp of logarithmic identities can greatly enhance one's ability to efficiently tackle logarithmic equations and expressions.

Properties of Logarithms

Properties of logarithms are essential to deconstructing and manipulating logarithmic expressions. These properties provide rules and shortcuts to simplify calculations.

• The power rule states that \(\log_{b}(a^n) = n \cdot \log_{b}{a},\) which allows us to pull down exponents as multiples.
• In the given exercise, recognizing that \(3^3 = 27\) and \(3^2 = 9\) showed us how to express \(\log_3{27}\) as 3 and \(\log_3{9}\) as 2, making subtraction straightforward.
• Another useful identity is the reciprocity rule: \[\log_b{a} = \frac{1}{\log_a{b}},\] which provides the insight to switch a logarithm's base and argument reciprocally.

Understanding these properties ensures a deeper comprehension of logarithmic operations and facilitates simpler arithmetic manipulations.

Simplification of Expressions

Simplifying expressions is often the ultimate goal in solving mathematical problems involving logarithms. It involves reducing an expression to its simplest form without changing its value.

• After applying the change of base formula and using the properties of logarithms to calculate specific outcomes, our focus shifts to arithmetic simplification.
• The expression \(\frac{3}{2} - \frac{2}{3}\) involves rational numbers, and simplifying such expressions requires finding a common denominator.
• In this case, the common denominator for 2 and 3 is 6. Rewriting the fractions with this common denominator gives \(\frac{9}{6} - \frac{4}{6} = \frac{5}{6}.\)

The result, \(\frac{5}{6},\) exemplifies the process of expression simplification: using mathematical identities, calculating individual values, and performing careful arithmetic operations. Emphasizing practice in these steps can solidify one's skills in simplifying complex expressions.