Problem 289 - Precalculus

Question

Prove that \(\log _{b}(n)=\frac{1}{\log _{n}(b)}\) for any positive integers \(b > 1\) and \(n > 1\)

Step-by-Step Solution

Answer

\(\log_b(n)\) and \(\log_n(b)\) are reciprocals as shown by the change of base formula.

Step 1: Understand the Relationship

To solve this, we need to understand the relationship between these two expressions. The expression \(\log_b(n)\) is the logarithm of \(n\) with base \(b\), and \(\log_n(b)\) is the logarithm of \(b\) with base \(n\). The goal is to prove that these two are reciprocals.

Step 2: Use the Change of Base Formula

The change of base formula is given by \(\log_b(n) = \frac{\log_k(n)}{\log_k(b)}\) for any positive base \(k\). We can use this to express both \(\log_b(n)\) and \(\log_n(b)\) in terms of a common base, usually 10 or \(e\).

Step 3: Apply to Both Logarithms

Choose a common base 10 for simplicity. Then, express \(\log_b(n)\) as \(\frac{\log_{10}(n)}{\log_{10}(b)}\) and \(\log_n(b)\) as \(\frac{\log_{10}(b)}{\log_{10}(n)}\). Now, the expression for \(\log_n(b)\) is seen as the reciprocal of \(\log_b(n)\).

Step 4: Conclude the Proof

Since \(\log_b(n) = \frac{\log_{10}(n)}{\log_{10}(b)}\) and \(\log_n(b) = \frac{\log_{10}(b)}{\log_{10}(n)}\), it is clear that \(\log_b(n) = \frac{1}{\log_n(b)}\) because one is the reciprocal of the other.

Key Concepts

Reciprocal Property of Logarithms

The reciprocal property of logarithms is an intriguing and essential concept in understanding logarithmic functions. This property states that for any two positive numbers, say \( b \) and \( n \) (where both are greater than 1), the logarithm of \( n \) with base \( b \) is the reciprocal of the logarithm of \( b \) with base \( n \). Simply put, this is expressed as:

• \( \log_{b}(n) = \frac{1}{\log_{n}(b)} \)

At first glance, this might seem abstract, but it’s truly about how exponentiation in different bases relates to division when looking at inverse operations in logarithms. This relationship hinges on how multiplication and division (or exponentiation and taking roots) are inverse operations. Hence, when you swap the base with the number and take the logarithm, you flip—or take the reciprocal of—the result.

Change of Base Formula

The change of base formula is a handy tool for converting logarithms from one base to another. Frequently, we use the formula to convert logarithms to a common base that's easier to work with, such as base 10 or base \( e \) (the natural logarithm base). This formula is:

• \( \log_b(n) = \frac{\log_k(n)}{\log_k(b)} \)

Here, \( k \) is any positive number, often 10 or \( e \), depending on which is convenient for calculation. This formula allows you to perform calculations with logarithms even when dealing with unusual or complex bases by using bases that have more straightforward operations or representations. For example, using common logs (base 10) or natural logs (base \( e \)) gives access to calculator functions, helping simplify the manual calculation process and proving reciprocal properties conveniently.

Logarithms with Different Bases

Understanding logarithms with different bases expands your mathematical toolkit, especially when working with exponential trends and growth patterns in various disciplines. Logarithms can have any positive number greater than 1 as a base, which changes how we interpret the power relationship between the input value and its logarithm.

• The base determines what power the base needs to be raised to yield the input number. For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).

The beauty of different bases is in tackling problems in unique contexts, such as binary systems (base 2), decimal systems (base 10), or natural logarithms \( \ln \) when using base \( e \). This facility aids in varied applications ranging from computer science to natural sciences, each having its suitable base aligning with its respective field’s demands—all underscoring the universality and flexibility of logarithmic functions.