Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Approximate the values of the logarithms by giving two consecutive integers, one of which is a lower bound and the other an upper bound for the expressions given. Do this without a calculator. (You can use the calculator to check your answers, but the idea of the problem is to get you to think about what logarithms mean.) Explain your reasoning as in the example below. (a) \(\log _{7} 50\) (b) \(\log _{10}(0.5)\)

Step-by-Step Solution

Answer

The lower and upper bounds for \( \log _{7} 50 \) are 1 and 2, respectively. The lower and upper bounds for \( \log _{10}(0.5) \) are -1 and 0, respectively.

Step 1: Understanding Logarithms

First, remember the mathematical definition of a logarithm: The logarithm base a of a number x, denoted as \( \log_a(x) \), is the exponent to which a must be raised to get x. In other words, if \( a^y = x \), then \( \log_a(x) = y \). Considering this, use the range of \( a^{n-1} < x < a^n \) to find the values of \( n-1 \) and \( n \) which will be the two consecutive integers representing the lower and upper bounds respectively.

Step 2: Solving for \( \log _{7} 50 \)

Given that 7 to the power of 1 is 7, and 7 to the power of 2 is 49, it can be deduced that \( \log _{7} 50 \) lies between 1 and 2. Hence, the lower and upper bounds of \( \log _{7} 50 \) are 1 and 2.

Step 3: Solving for \( \log _{10}(0.5) \)

Given that 10 to the power of 0 is 1, and 10 to the power of -1 is 0.1, it can be deduced that \( \log _{10}(0.5) \) lies between 0 and -1. Hence, the lower and upper bounds of \( \log _{10}(0.5) \) are -1 and 0.

Key Concepts

Upper and Lower Bounds

Upper and lower bounds help us to understand the range where a certain value lies. This concept is widely used in mathematics to approximate values, especially when exact computation is not possible or practical. In the context of logarithms, these bounds give us two consecutive integers between which the logarithm value falls.

For instance, when finding the logarithm of a number, we look for two exponents of the base number that bracket the target value.

• The lower bound is where the base raised to that power is less than the given number.
• The upper bound is where the base raised to that power is more than the given number.

By establishing these bounds, you get a clearer picture of where the logarithm value lies, even without exact calculations. This not only enhances your estimation skills but also deepens your understanding of logarithmic operations.

Mathematical Reasoning

Mathematical reasoning is essential in evaluating and understanding problems involving logarithms. It requires breaking down a problem into understandable segments and applying logical steps to reach a solution.

To approximate the values of logarithms, as seen in the original exercise, we need to:

• Understand the properties of logarithms and exponentials.
• Estimate values by comparing them to known numbers or exponents.
• Use logical deduction to establish the upper and lower bounds.

For example, knowing that for 7 to be raised to which power leads closely to 50, you explore the powers of 7 that are close to 50. Observing that 7¹ = 7 and 7² = 49, you can logically deduce that the logarithm of 50 base 7 lies between those two exponents.

With targeted reasoning, we apply our knowledge not through blind calculation but through understanding relationships between numbers.

Integers

Integers are a fundamental concept in mathematics, essential for understanding the bounds we calculate in logarithms. An integer is simply a whole number that can be positive, negative, or zero. They are particularly important when it comes to representing the exponents in logarithmic equations.

When estimating a logarithmic value, we often determine which consecutive integers its value lies between. These integers, as upper and lower bounds, do not have fractions or decimals, making them easy to work with when calculating potential approximations.

• Lower bound = smaller integer, where the exponent result is less than the target value.
• Upper bound = larger integer, where the exponent result is more than the target value.

For example, when estimating \(\log_7 (50)\), we find it between 1 and 2, because 7 raised to the power of 1 is 7, and to the power of 2 is 49, both integers.

By working with integers, you ensure simplicity and clarity in understanding your calculations and results.