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Chapter 5: Problem 73

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

A learning curve is a graph of a function \(P(t)\) that measures the performance of someone learning a skill as a function of the training time \(t\) . At first, the rate of learning is rapid. Then, as performance increases and approaches a maximal value \(M\) , the rate of learning decreases. It has been found that the function $$P(t)=M-C e^{-k t}$$ where \(k\) and \(C\) are positive constants and \(C< M\) is a reasonable model for learning. (a) Express the learning time \(t\) as a function of the performance level \(P .\) (b) For a pole-vaulter in training, the learning curve is given by $$P(t)=20-14 e^{-0.024 t}$$ where \(P(t)\) is the height he is able to pole-vault after \(t\) months. After how many months of training is he able to vault 12 \(\mathrm{ft}\) ? (c) Draw a graph of the learning curve in part (b).

\(29-43\) . These exercises deal with logarithmic scales. Earthquake Magnitudes If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

Squirrel Population A grey squirrel population was in- troduced in a certain county of Great Britain 30 years ago. Biologists observe that the population doubles every 6 years, and now the population is \(100,000\) . (a) What was the initial size of the squirrel population? (b) Estimate the squirrel population 10 years from now. (c) Sketch a graph of the squirrel population.

\(25-28=\) These exercises use Newton's Law of Cooling. Cooling Turkey A roasted turkey is taken from an oven when its temperature has reached \(185^{\circ} \mathrm{F}\) and is placed on a table in a room where the temperature is \(75^{\circ} \mathrm{F} .\) (a) If the temperature of the turkey is \(150^{\circ} \mathrm{F}\) after half an hour, what is its temperature after 45 \(\mathrm{min}\) ? (b) When will the turkey cool to \(100^{\circ} \mathrm{F} ?\)

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$ f(x)=2^{x}, \quad g(x)=x+1 $$
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