Problem 73 - Algebra and Trigonometry

Question

Find the Error What is wrong with the following argument? $$ \begin{aligned} \log 0.1 &<2 \log 0.1 \\ &=\log (0.1)^{2} \\ &=\log 0.01 \\\ \log 0.1 &<\log 0.01 \\ 0.1 &<0.01 \end{aligned} $$

Step-by-Step Solution

Answer

The error involves incorrectly comparing \( \log 0.1 \) with \( \log 0.01 \), leading to a false inequality \( 0.1 < 0.01 \).

Step 1: Understand Logarithmic Properties

The problem involves operations with logarithms, so it's important to review relevant properties. Particularly, remember that \( \log(a^b) = b \log(a) \) and logarithms are monotonically increasing for base greater than 1.

Step 2: Evaluate Properties Applied

In the expression \( 2 \log 0.1 = \log (0.1)^2 \), property \( \log(a^b) = b \log(a) \) is applied correctly. However, the step involves comparing \( \log 0.1 \) and \( \log 0.01 \). Evaluate whether the steps use the properties correctly and whether any incorrect conclusions are drawn.

Step 3: Identify Logical Errors

The mistake lies in assuming that \( x < 2x \) implies \( \log x < \log(0.01) \), without considering the fact that \( \log(x) < \log(y) \) only when \( x > y \) given the logarithm properties. Instead, if based upon the structure, \( \log(x) < \log(x^2) \), this incorrectly influenced a wrong relation: \( x < x^2 \) inherently assumes incorrect inequality.Thus, the conclusion \( \log 0.1 < \log 0.01 \) leads to \( 0.1 < 0.01 \), which is false.

Key Concepts

Logarithm Rules

Logarithm rules play a crucial role when working with logarithms in mathematics. These rules help simplify expressions and solve logarithmic equations. Here are some of the key logarithm rules:

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

These rules are handy for manipulating and working with logarithmic expressions. They allow us to transform complex log problems into simpler forms. In the context of the original problem, the Power Rule was used to transform \( 2 \log 0.1 = \log (0.1)^2 \), illustrating a correct application of the rule.

Monotonicity of Logarithms

Monotonicity refers to the behavior of a function regarding its slope - whether it consistently increases or decreases. For logarithms, this property is dependent on the base of the logarithm:

• If the base \( b > 1 \), then the logarithmic function is monotonically increasing. This means that if \( x > y \), then \( \log_b(x) > \log_b(y) \).
• If the base \( 0 < b < 1 \), then the logarithmic function is monotonically decreasing. In this scenario, if \( x > y \), then \( \log_b(x) < \log_b(y) \).

In our example, common logarithms with base 10 are implicitly considered, which are monotonically increasing. This means if you compare \( \log(0.1) \) and \( \log(0.01) \) under this property, \( \log(0.1) > \log(0.01) \) since \( 0.1 > 0.01 \). Hence, the assumption made in the problem that \( \log 0.1 < \log 0.01 \) was incorrect.

Common Logarithm Errors

When solving logarithm problems, students often make certain common errors. Being aware of these can help you avoid them:

• Assuming Additivity: Some may assume \( \log(x + y) = \log(x) + \log(y) \), which is not true. The correct form is the Product Rule: \( \log(xy) = \log(x) + \log(y) \).
• Ignoring Base Influence: Forgetting about the base’s impact on the monotonicity can lead to incorrect conclusions about order and size.
• Incorrect Inequality Interpretations: If \( \log(x) < \log(y) \), it implies \( x < y \) only if the logarithmic function is increasing (base > 1). Misinterpreting this, as seen in the original exercise, leads to faulty results.
• Misapplying Rules: Misapplication of the Power Rule, such as wrongly distributing exponents, also leads to errors.

Understanding these errors and reviewing the basic properties of logarithms help deepen comprehension and enhance accuracy in solving logarithmic equations.