Problem 21 - Technical Mathematics with Calculus

Question

Find the common logarithm of each number. $$27.4$$

Step-by-Step Solution

Answer

The common logarithm of 27.4 is approximately 1.4378.

Step 1: Identify the number for logarithm

Read the problem carefully to identify the number for which the common logarithm is to be found. In this case, the number is 27.4.

Step 2: Use calculator or logarithm table

Use a calculator or a logarithm table to find the common logarithm of 27.4. The common logarithm is the logarithm with base 10, often written as \(\log_{10}\) or simply \(\log\).

Step 3: Calculate or lookup the common logarithm

When using a calculator, simply enter 27.4 and press the log function to get the result. If you are using a logarithm table, find the row for 27 and then the column for 4 to find the common logarithm of 27.4.

Step 4: Write down the answer

The common logarithm of 27.4 is approximately 1.4378. This value is subject to rounding depending on the precision of your calculator or the logarithm table.

Key Concepts

Logarithm Table

Understanding the usage of a logarithm table is essential for performing logarithmic calculations when a calculator is not at hand.

A logarithm table lists the logarithms of numbers, usually in base 10, which is known as the common logarithm. It arranges numbers in a systematic grid where one can find the logarithm of a number by matching its mantissa (the main part of the number) and characteristic (the integer part of the logarithm). For instance, to find the common logarithm of 27.4 using a table, locate the value corresponding to 27 (the main part) and then slide over to the column for the decimal part, which is .4 in this case. The value found there must then be adjusted by adding the characteristic, typically found in the leftmost column.

Utilizing a logarithm table effectively requires familiarity with interpreting these values and the layout of the table itself. With practice, you can quickly find logarithms of various numbers using this manual method, making it a valuable tool in mathematical problem solving, particularly when technology is unavailable.

Logarithmic Calculations

Logarithmic calculations are fundamental in various fields of science, engineering, and finance. A common logarithm, as seen in our example with the number 27.4, is specifically the logarithm with base 10. It is represented as \( \log_{10}(27.4) \) or simply \( \log(27.4) \).

When using a calculator to find the common logarithm, the process is straightforward: input the number and press the logarithm function key, denoted as 'log'. Calculators provide the advantage of quick and accurate results, suitable for complex numbers where a logarithm table might not suffice.

But understanding the calculation process is crucial for deeper mathematical comprehension. It's about recognizing that logarithms answer the question of how many of one number (the base) we need to multiply to get another number. In educational settings and some practical scenarios, knowing how to read logarithmic tables and perform manual calculations enhances one's capability to solve problems effectively and fosters a stronger grasp of the concept.

Mathematical Problem Solving

Mathematical problem solving involves a step-by-step approach to tackle complex mathematical questions. The steps demonstrated in finding the common logarithm of 27.4 exemplify such an approach. Initially, one identifies the given number before selecting the appropriate tool, such as a logarithm table or calculator, for the calculation needed.

Every problem-solving process includes:

• Understanding the problem,
• Devising a plan,
• Carrying out that plan,
• Reviewing the solution for accuracy.

The ability to strategically approach problems and use available tools efficiently, such as knowing when to opt for a calculator over a logarithm table, is an essential skill in mathematics. Additionally, verifying the final result and understanding the implications of approximations or rounding errors are crucial for accurate and reliable solutions.

Exercises like finding common logarithms are designed to improve mathematical thinking and problem-solving skills, which are vital across all levels of math education and beyond.