Problem 41 - College Algebra with Corequisite Support

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

The equality holds due to the change of base formula.

Step 1: Understand the Equality You Need to Prove

We want to prove that \(\log _{b}(n)=\frac{1}{\log _{n}(b)}\). On one side we have the logarithm of \(n\) with base \(b\), and on the other side we have the reciprocal of the logarithm of \(b\) with base \(n\). Our task is to show that these two expressions are equal.

Step 2: Introduce Change of Base Formula

The change of base formula for logarithms states that \(\log_{a}(c) = \frac{\log_{k}(c)}{\log_{k}(a)}\) for any positive base \(k\) different from 1. This formula allows us to express any logarithm in terms of logarithms of another base.

Step 3: Apply Change of Base to \(\log_{b}(n)\)

Using the change of base formula, we express \(\log_{b}(n)\) in terms of base 10 (common logarithm): \(\log_{b}(n) = \frac{\log_{10}(n)}{\log_{10}(b)}\).

Step 4: Apply Change of Base to \(\log_{n}(b)\)

Similarly, express \(\log_{n}(b)\) using the change of base formula: \(\log_{n}(b) = \frac{\log_{10}(b)}{\log_{10}(n)}\).

Step 5: Take the Reciprocal of \(\log_{n}(b)\)

Since we want \(\frac{1}{\log_{n}(b)}\), take the reciprocal of the expression we found in Step 4: \(\frac{1}{\log_{n}(b)} = \frac{\log_{10}(n)}{\log_{10}(b)}\).

Step 6: Compare and Conclude

Notice that the expressions \(\log_{b}(n)\) and \(\frac{1}{\log_{n}(b)}\) are both equal to \(\frac{\log_{10}(n)}{\log_{10}(b)}\). Therefore, we have shown that \(\log_{b}(n) = \frac{1}{\log_{n}(b)}\).

Key Concepts

Change of Base Formula

The Change of Base Formula is a powerful tool in logarithms, which helps us calculate logarithms using different bases. It's especially handy when dealing with bases that are not commonly used, like base 10 (common logarithm) or base 2 (binary logarithm). The formula is as follows:

• \( \log_{a}(c) = \frac{\log_{k}(c)}{\log_{k}(a)} \)

Here, \(a\) is the base you want to convert from, \(c\) is the number you are taking the log of, and \(k\) is the new base you are converting to. This formula helps express \(\log_{a}(c)\) in terms of logarithms of another base, making complex calculations more manageable.
For example, if you need to calculate \(\log_{b}(n)\), but your calculator only handles base 10 logarithms, the change of base formula lets you convert it to:

• \(\log_{b}(n) = \frac{\log_{10}(n)}{\log_{10}(b)} \)

Using this method makes it easier to handle logarithmic calculations without needing specialized tools for various bases.

Reciprocal Property of Logarithms

The Reciprocal Property of Logarithms is an interesting mathematical identity, which shows the relationship between two logarithms of the same arguments but with reciprocal bases. It states that:

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

This means that if you have the logarithm of a number \(n\) in base \(b\), it's equal to the reciprocal of the logarithm of \(b\) in base \(n\).
This property can be very useful. For instance, when you need to switch between perspectives, like viewing a power in one base versus another. Understanding this property simplifies certain mathematical proofs and calculations by demonstrating this reciprocal relationship between log functions, turning perceived challenges into manageable tasks.

Mathematical Proofs

Mathematical proofs form the backbone of mathematics. They provide a way to demonstrate that a particular statement or phenomenon is true. A proof is a logical argument that uses previously established facts, definitions, and properties to show the truth of a statement.
When dealing with logarithms, as seen in the original exercise, proofs establish the validity of identities like the Reciprocal Property. The steps go as follows:

• Starting with a known truth or assumption (in this case, the Change of Base Formula).
• Manipulate the equations step by step, maintaining equality throughout.
• Use logical equivalences (like taking reciprocals) to reach the desired conclusion.

The beauty of a mathematical proof is in its clarity and lack of ambiguity. Once a proof is established, it holds universally, providing a solid foundation upon which other knowledge can be built.