Problem 63 - Algebra 2

Question

Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. For all positive numbers \(x\) and \(y, \frac{\log x}{\log y}=\frac{\ln x}{\ln y}\)

Step-by-Step Solution

Answer

The statement is always true.

Step 1: Understand the Problem

We need to determine whether the given expression \(\frac{\log x}{\log y} = \frac{\ln x}{\ln y}\) is sometimes, always, or never true for all positive numbers \(x\) and \(y\). The expression involves two different logarithmic bases: base 10 (\(\log\)) and base \(e\) (\(\ln\)).

Step 2: Review Logarithmic Properties

Recall that for a logarithm of any base \(b\), the change of base formula can be defined as: \[ \log_b a = \frac{\log_k a}{\log_k b} \]where \(k\) can be any positive base different from 1. Using this, both \(\log\) and \(\ln\) can be expressed in terms of each other or a common base.

Step 3: Apply Change of Base Formula

Apply the change of base formula for \(\log x\) and \(\log y\) using the natural logarithm (\(\ln\)) as their base:\[ \log x = \frac{\ln x}{\ln 10} \] \[ \log y = \frac{\ln y}{\ln 10} \]Thus, the expression \(\frac{\log x}{\log y}\) becomes:\[ \frac{\ln x / \ln 10}{\ln y / \ln 10} = \frac{\ln x}{\ln y} \]The \(\ln 10\) terms cancel out.

Step 4: Conclusion

Since after simplification, \(\frac{\log x}{\log y}\) simplifies to \(\frac{\ln x}{\ln y}\), the expression \(\frac{\log x}{\log y} = \frac{\ln x}{\ln y}\) holds for all positive numbers \(x\) and \(y\). Hence, it is **always true.**

Key Concepts

Change of Base Formula

The change of base formula is a crucial tool in logarithmic calculations. This formula allows you to convert a logarithm from one base to another. This can be especially helpful when dealing with different bases, like base 10, represented as \(\log\), and base \(e\), represented as \(\ln\).

The formula is expressed as:
\[\log_b a = \frac{\log_k a}{\log_k b} \]
where \(b\) is the original base, and \(k\) is the new base we are converting to. This change is possible because logarithms can be expressed in terms of **any** base, as long as that base is positive and not equal to one.

Applying the change of base formula allows us to convert between logarithms with different bases, making computations easier and more flexible. In our exercise, we used the natural logarithm as the base to demonstrate that the expression \(\frac{\log x}{\log y} = \frac{\ln x}{\ln y}\) is always true, regardless of the values of \(x\) and \(y\) as long as they are positive.

Natural Logarithm

The natural logarithm, denoted \(\ln\), is a specific logarithm with the base of \(e\), an irrational number approximately equal to 2.718. The natural logarithm is widely used in mathematics due to its unique properties and simplicity in differentiation and integration.

For any positive number \(x\), \(\ln x\) is defined as the power to which \(e\) must be raised to obtain \(x\). It is commonly used in calculus, particularly in dealing with exponential growth and decay problems, because of its natural properties that simplify mathematical analysis.

• The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\).
• The natural logarithm of 1 is 0. That is, \(\ln 1 = 0\).
• \(\ln e = 1\) since \(e^1 = e\).

These properties make \(\ln\) a useful and powerful tool in various branches of mathematics and sciences.

Logarithmic Properties

Logarithms have a set of properties that make them incredibly useful for solving complex problems. These properties allow for simplification and transformation of expressions, which is handy in both theoretical and applied mathematics.

The main logarithmic properties include:

• **Product Property**: \(\log_b(xy) = \log_b x + \log_b y\)
• **Quotient Property**: \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• **Power Property**: \(\log_b(x^k) = k\log_b x\)
• **Change of Base Formula**: \(\log_b a = \frac{\log_k a}{\log_k b}\)

These properties facilitate the manipulation of logarithmic expressions and equations.

In our exercise, we effectively used the change of base formula, combining it with the property that allows division inside a logarithm to be expressed as a difference. It showed \(\frac{\log x}{\log y}\) and \(\frac{\ln x}{\ln y}\) reduce to the same expression, proving it is always true. Understanding these properties ensures a strong foundational grasp on handling problems involving logarithms.