Problem 91 - College Algebra and Calculus: An Applied Approach

Question

Demonstrate that \(\frac{\ln x}{\ln y} \neq \ln \frac{x}{y}=\ln x-\ln y\) by using a spreadsheet to complete the table. $$ \begin{array}{|c|c|c|c|c|} \hline x & y & \frac{\ln x}{\ln y} & \ln \frac{x}{y} & \ln x-\ln y \\ \hline 1 & 2 & & & \\ \hline 3 & 4 & & & \\ \hline 10 & 5 & & & \\ \hline 4 & 0.5 & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The values of \(\frac{\ln x}{\ln y}\) and \(\ln x - \ln y\) are not equivalent and hence the statement is proven.

Step 1: Understanding the Values to be Substituted

Take the \(x\) and \(y\) values from the table given in the question to be substituted into all three expressions \(\frac{\ln x}{\ln y}\), \(\ln \frac{x}{y}\) and \(\ln x - \ln y\). The first column represents \(x\) values and the second column represents \(y\) values. The \(x\) and \(y\) pairs are (1,2), (3,4), (10,5) and (4,0.5).

Step 2: Calculate the Expressions for Each Pair

Substitute each pair of \(x\) and \(y\) values into the three expressions and calculate their results. Ensure to handle natural logarithms properly as they involve base e in their calculations.

Step 3: Fill the Table

Once all the results have been calculated, fill in the table. The results for \(\frac{\ln x}{\ln y}\) go in the third column, the results for \(\ln \frac{x}{y}\) go in the fourth column, and the results for \(\ln x - \ln y\) go in the fifth column.

Step 4: Compare the Results

Upon comparison, observe whether the values of \(\frac{\ln x}{\ln y}\) and \(\ln x - \ln y\) are equivalent for the same \(x\) and \(y\) pair. If the values are different, the given statement is proven.

Key Concepts

Logarithmic Identities

Logarithmic identities are the backbone of simplifying complex logarithmic expressions and solving equations involving logarithms. An understanding of these identities helps in accurately navigating mathematical problems that feature logarithms. One of the most fundamental identities states that the logarithm of a quotient is equal to the difference of the logarithms, specifically ewline \[ \ln \frac{x}{y} = \ln x - \ln y \].ewline This identity applies to natural logarithms, which use the base \(e\), a mathematical constant. Another common identity is the logarithm of a product, ewline \[ \ln(xy) = \ln(x) + \ln(y) \].ewline An understanding of these allows students to reframe logarithmic expressions in various forms for easier computation and analysis.

Mathematical Proof

Mathematical proof serves as definitive evidence that a mathematical statement is true. Proofs often use a logical sequence of statements, starting from known facts and using accepted mathematical operations to arrive at the conclusion. The exercise involves proving that ewline \[ \frac{\ln x}{\ln y} eq \ln \frac{x}{y} \] ewline by substituting values and making comparisons. While this isn't a formal proof, it's an illustrative method to show that the properties of logarithms do not support this equation for all values of \(x\) and \(y\). A more formal proof would involve algebraic manipulation and use of known logarithmic identities to demonstrate why the two expressions cannot be equivalent.

Algebraic Manipulation

Algebraic manipulation involves the rearrangement and simplification of algebraic expressions and equations. It's a crucial skill in mathematics that enables the direct comparison of quantities or solving equations. In relation to logarithmic expressions, algebraic manipulation may involve applying logarithmic identities to simplify or expand expressions. For instance, the expression ewline \[ \ln(x^2) \] ewline can be simplified using logarithmic identities to ewline \[ 2 \ln(x) \].ewline In the exercise provided, algebraic manipulation is used to compute logarithmic expressions based on given \(x\) and \(y\) values to compare and demonstrate the inequality between different logarithmic forms.

Logarithmic Expressions

Logarithmic expressions involve the logarithm function, commonly the natural logarithm (\(ln\)) which has the base \(e\). Logarithms are used to understand the power to which a number must be raised to achieve another number, embodying the inverse of exponentiation. For example, ewline \[ \ln(e^x) = x \] ewline reflects that \(e\) raised to the power of \(x\) gives \(e^x\), of which \(x\) is the natural logarithm. Comparing logarithmic expressions often requires careful consideration of their properties and domains. In the exercise, we're looking at different logarithmic forms based on the quotient of numbers, fostering a nuanced understanding of how natural logarithms behave with different operations.