Problem 112 - Elementary and Intermediate Algebra

Question

How can the LOG key on a calculator be used to find \(\log _{2} 7 ?\)

Step-by-Step Solution

Answer

To find \( \log_2 7 \), use the formula \( \log_2 7 = \frac{\log_{10} 7}{\log_{10} 2} \) and calculate using a calculator's log function.

Step 1: Understand the Goal

We need to find the value of \( \log_2 7 \), which is the logarithm of 7 with base 2. Most calculators have a \( \log \) key that calculates logarithms with base 10 (common logarithm) and sometimes a \( \ln \) key for base \( e \). To find logarithms with a different base, we must use the change of base formula.

Step 2: Apply the Change of Base Formula

The change of base formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a base that the calculator can handle, usually 10 or \( e \). So, \( \log_2 7 = \frac{\log_{10} 7}{\log_{10} 2} \).

Step 3: Calculate \( \log_{10} 7 \)

Use the calculator's \( \log \) key. Enter 7, then press \( \log \) to find \( \log_{10} 7 \). Make sure to note this value for later use.

Step 4: Calculate \( \log_{10} 2 \)

Similarly, use the calculator's \( \log \) key. Enter 2, then press \( \log \) to find \( \log_{10} 2 \). This value is important for dividing in the next step.

Step 5: Compute \( \log_2 7 \) Using the Values

Take the value from Step 3 for \( \log_{10} 7 \) and divide it by the value from Step 4 for \( \log_{10} 2 \). This gives you \( \log_2 7 \). Your calculator will handle the arithmetic and provide an approximate value.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics. They answer the question: "To what exponent must a certain base be raised, in order to obtain a specific number?" For example, if we say \( \log_2 8 \), we are essentially asking, "What power must 2 be raised to in order to get 8?" In this case, the answer is 3, because \( 2^3 = 8 \).
Logarithms have a wide range of applications, from solving equations in algebra to analyzing exponential growth in various sciences. They also help to reduce multiplication into addition when dealing with large numbers, a technique that was crucial before the era of digital calculators. This mathematical tool can be expressed in various bases, and most commonly, base 10 (known as common logarithm) and base \(e\) (natural logarithm) are used.
Key points about logarithms:

• The logarithm base is the number that is repeatedly multiplied.
• Base 10 logarithms are common in real-life applications.
• Logarithms can "undo" the process of exponentiation.

Calculator Usage

Modern calculators are equipped with functionalities to simplify complex math calculations. Most of them come with specific keys labeled as \(\log\) and \(\ln\), which stand for the common and natural logarithm functions, respectively. For a student, understanding how to perform tasks like computing logarithms using a calculator is essential, especially when they deal with different bases like in this exercise.
To use a calculator for finding the logarithm of a given number:

• Ensure the calculator is in the correct mode for your calculations.
• Identify and press the \(\log\) key to find the common logarithm of a number using base 10.
• For natural logarithms, use the \(\ln\) key.

When dealing with bases that are not readily available on calculators, use the change of base formula to convert the logarithmic operation into one that the calculator can process.

Common Logarithm

A common logarithm has a base of 10 and is denoted simply as \(\log\). This type of logarithm is extensively used in calculations involving scientific and engineering contexts. Because calculators typically default to base 10 for the \(\log\) function, it is crucial to be familiar with the idea.
Common log operation enables us to break down complex multiplication into simpler addition problems, thanks to their fundamental properties, such as:

• \(\log_{10}(ab) = \log_{10}(a) + \log_{10}(b)\)
• \(\log_{10}(\frac{a}{b}) = \log_{10}(a) - \log_{10}(b)\)
• \(\log_{10}(a^n) = n \cdot \log_{10}(a)\)

Understanding these properties and how to use the common logarithm key on a calculator helps you solve various math problems efficiently and accurately.

Mathematics Education

Understanding and applying logarithms are key aspects of mathematics education. This concept not only appears in higher mathematics but is also a part of basic education due to its applications across multiple disciplines. Students progress from understanding basic exponents to manipulating and computing logarithms.
Teaching logarithms effectively involves:

• Introducing the idea of inverse functions early in the math curriculum.
• Utilizing practical examples, such as sound decibels and the Richter scale, to demonstrate real-world application.
• Using digital tools like calculators to instill proficiency in handling logarithms with different bases.

The goal of mathematics education is to equip students with the necessary problem-solving skills. Through examples and regular practice, students become adept at using mathematical concepts such as logarithms in everyday situations and academic pursuits.