Evaluating Expressions Using Log Tables And Standard Form
Introduction
In mathematics, we often encounter complex calculations involving multiplication, division, and roots. Logarithm tables, also known as log tables, provide a convenient method to simplify these calculations, especially when dealing with large numbers or intricate expressions. This article will guide you through the process of using log tables to evaluate expressions and express the answer in standard form. We will use the example expression:
√(853 * 4) / (54 * 81)
This expression involves multiplication, division, and a square root, making it a perfect candidate for log table evaluation. Standard form, also known as scientific notation, is a way of expressing numbers as a decimal number between 1 and 10 multiplied by a power of 10. It's particularly useful for representing very large or very small numbers concisely. To effectively utilize log tables, it's essential to understand the fundamental principles of logarithms and their properties. Logarithms are essentially the inverse operation of exponentiation. If we have an equation like aˣ = y, the logarithm of y to the base a is x, written as logₐ(y) = x. The most commonly used logarithm is the common logarithm, which has a base of 10, denoted as log₁₀(y) or simply log(y). The properties of logarithms that make them useful for calculations include: log(a * b) = log(a) + log(b), log(a / b) = log(a) - log(b), and log(aⁿ) = n * log(a). These properties allow us to convert multiplication and division into addition and subtraction, and exponentiation into multiplication, which are easier to handle using log tables. The log table typically consists of two main parts: the characteristic and the mantissa. The characteristic is the integer part of the logarithm and indicates the power of 10, while the mantissa is the decimal part and is found in the log table. Understanding these concepts and how to use the log table is crucial for accurate calculations. We will delve deeper into the steps of using log tables and standard form to solve the given expression in the following sections.
Step 1: Apply Logarithms to the Expression
To begin, we apply logarithms to the entire expression. This step is crucial as it transforms complex operations like multiplication, division, and square roots into simpler addition, subtraction, and multiplication, respectively. By taking the logarithm of both sides, we can leverage the properties of logarithms to break down the expression into manageable parts. The expression we are evaluating is:
√(853 * 4) / (54 * 81)
Let's denote the value of this expression as 'x'. So, we have:
x = √(853 * 4) / (54 * 81)
Now, we take the logarithm (base 10) of both sides:
log₁₀(x) = log₁₀[√(853 * 4) / (54 * 81)]
Using the properties of logarithms, we can simplify the expression on the right-hand side. Recall that the logarithm of a quotient is the difference of the logarithms, the logarithm of a product is the sum of the logarithms, and the logarithm of a number raised to a power is the power times the logarithm of the number. Applying these rules, we get:
log₁₀(x) = log₁₀[√(853 * 4)] - log₁₀(54 * 81)
We can further break down the square root and the products:
log₁₀(x) = log₁₀[(853 * 4)^(1/2)] - [log₁₀(54) + log₁₀(81)]
Applying the power rule of logarithms, we get:
log₁₀(x) = (1/2) * [log₁₀(853) + log₁₀(4)] - [log₁₀(54) + log₁₀(81)]
This equation now represents the original expression in terms of logarithms of individual numbers, which can be easily looked up in a log table. This step is vital as it sets the stage for using log tables to find the logarithmic values of each number, which we will do in the next step. By converting the expression into logarithmic form, we have transformed a complex calculation into a series of additions, subtractions, and multiplications that are far simpler to handle. This approach significantly reduces the risk of errors and makes the overall calculation more manageable. The ability to manipulate expressions using logarithmic properties is a fundamental skill in mathematics and is particularly useful in various fields such as engineering, physics, and computer science.
Step 2: Find the Logarithms Using Log Tables
In this crucial step, we utilize log tables to find the logarithmic values of the individual numbers in our expression. Log tables are indispensable tools for simplifying complex calculations, and understanding how to use them is fundamental to this process. Log tables typically provide the logarithms (base 10) of numbers between 1 and 10. To find the logarithm of any number, we need to express it in standard form, which is a number between 1 and 10 multiplied by a power of 10. For instance, 853 can be written as 8.53 * 10². The power of 10 (in this case, 2) gives us the characteristic of the logarithm, and the log table gives us the mantissa, which is the decimal part of the logarithm. Our expression from Step 1 is:
log₁₀(x) = (1/2) * [log₁₀(853) + log₁₀(4)] - [log₁₀(54) + log₁₀(81)]
We need to find the values of log₁₀(853), log₁₀(4), log₁₀(54), and log₁₀(81). Let's start with log₁₀(853). First, we express 853 in standard form: 853 = 8.53 * 10². The characteristic is 2. To find the mantissa, we look up 85 in the log table and find the value corresponding to the column 3. The mantissa is approximately 0.9309. Therefore, log₁₀(853) ≈ 2.9309. Next, we find log₁₀(4). In standard form, 4 is 4.0 * 10⁰. The characteristic is 0. We look up 40 in the log table (since 4 is the same as 4.0) and find the mantissa, which is approximately 0.6021. Thus, log₁₀(4) ≈ 0.6021. Now, let's find log₁₀(54). In standard form, 54 = 5.4 * 10¹. The characteristic is 1. We look up 54 in the log table and find the mantissa, which is approximately 0.7324. Therefore, log₁₀(54) ≈ 1.7324. Finally, we find log₁₀(81). In standard form, 81 = 8.1 * 10¹. The characteristic is 1. We look up 81 in the log table and find the mantissa, which is approximately 0.9085. Thus, log₁₀(81) ≈ 1.9085. In summary, we have:
log₁₀(853) ≈ 2.9309
log₁₀(4) ≈ 0.6021
log₁₀(54) ≈ 1.7324
log₁₀(81) ≈ 1.9085
These values are crucial for the next step, where we will substitute them back into the equation and simplify to find the value of log₁₀(x). By accurately finding the logarithmic values using the log tables, we ensure the precision of our final answer. The process of using log tables may seem tedious at first, but with practice, it becomes a quick and efficient method for handling complex calculations. This skill is particularly valuable in situations where calculators are not available or when dealing with very large or very small numbers.
Step 3: Substitute the Logarithm Values and Simplify
After finding the logarithms of the individual numbers using log tables, the next critical step is to substitute these values back into the equation derived in Step 1. This substitution allows us to transform the logarithmic expression into a numerical calculation, which we can then simplify using basic arithmetic operations. Our equation from Step 1 was:
log₁₀(x) = (1/2) * [log₁₀(853) + log₁₀(4)] - [log₁₀(54) + log₁₀(81)]
And from Step 2, we found the approximate logarithmic values:
log₁₀(853) ≈ 2.9309
log₁₀(4) ≈ 0.6021
log₁₀(54) ≈ 1.7324
log₁₀(81) ≈ 1.9085
Now, we substitute these values into the equation:
log₁₀(x) ≈ (1/2) * [2.9309 + 0.6021] - [1.7324 + 1.9085]
First, we perform the addition inside the brackets:
log₁₀(x) ≈ (1/2) * [3.5330] - [3.6409]
Next, we multiply 3.5330 by 1/2:
log₁₀(x) ≈ 1.7665 - 3.6409
Now, we subtract 3.6409 from 1.7665:
log₁₀(x) ≈ -1.8744
This result gives us the logarithm of x. The negative value indicates that the number x is less than 1. This is a crucial intermediate result. To find the value of x, we need to take the antilogarithm of -1.8744, which will be covered in the next step. The process of substituting the logarithmic values and simplifying the equation is essential for reducing the complexity of the calculation. By breaking down the problem into smaller, manageable steps, we minimize the chances of making errors. This methodical approach is a key principle in mathematics and is applicable to a wide range of problems. The ability to perform arithmetic operations with logarithmic values is a fundamental skill that builds upon the understanding of logarithmic properties and the use of log tables.
Step 4: Find the Antilogarithm
After simplifying the equation and obtaining the value of log₁₀(x), the next essential step is to find the antilogarithm. The antilogarithm, also known as the inverse logarithm, is the process of finding the number whose logarithm is known. In our case, we have found that log₁₀(x) ≈ -1.8744, and we need to find the value of x. To find the antilogarithm of a negative number, it's helpful to express the logarithm in a form where the decimal part (mantissa) is positive. We can do this by adding and subtracting an integer value. In this case, we add and subtract 2:
log₁₀(x) ≈ -1.8744 = -2 + (2 - 1.8744)
log₁₀(x) ≈ -2 + 0.1256
Now we have the logarithm in the form -2 + 0.1256, where -2 is the characteristic and 0.1256 is the positive mantissa. The antilogarithm table is used to find the number corresponding to the mantissa. We look up 0.1256 in the antilogarithm table. The value corresponding to 0.1256 is approximately 1.335. This value represents the digits of our number. The characteristic (-2) tells us the position of the decimal point. A characteristic of -2 means that the decimal point should be moved two places to the left. Therefore, the antilogarithm of -1.8744 is:
x ≈ 1.335 * 10⁻²
x ≈ 0.01335
This value is the result of our original expression. The process of finding the antilogarithm involves using the antilogarithm table and correctly interpreting the characteristic to determine the decimal point's position. This step is crucial for converting the logarithmic value back into the original numerical value. The accuracy of the final result depends heavily on the precise use of the antilogarithm table and the correct handling of the characteristic. The ability to find antilogarithms is a fundamental skill in logarithmic calculations and is essential for solving various mathematical and scientific problems. By mastering this step, one can confidently convert logarithmic results back into their original form, making the entire process of using logarithms for calculations complete and effective.
Step 5: Express the Answer in Standard Form
After finding the value of x, the final step is to express the answer in standard form, also known as scientific notation. Standard form is a way of writing numbers as a product of a number between 1 and 10 and a power of 10. This notation is particularly useful for representing very large or very small numbers concisely and is widely used in scientific and engineering fields. From Step 4, we found that:
x ≈ 0.01335
To express this number in standard form, we need to rewrite it as a number between 1 and 10 multiplied by a power of 10. We move the decimal point two places to the right to get 1.335. Since we moved the decimal point two places to the right, we multiply by 10⁻² to compensate for this shift. Therefore, the standard form of 0.01335 is:
x ≈ 1.335 * 10⁻²
This is the final answer expressed in standard form. It is a concise and clear representation of the value of the original expression. Expressing the answer in standard form ensures consistency and facilitates easier comparison and manipulation of numbers, especially in scientific contexts. The process of converting a number to standard form involves identifying the digits between 1 and 10 and determining the appropriate power of 10. This skill is fundamental in mathematics and science and is crucial for effectively communicating numerical results. By expressing the final answer in standard form, we complete the evaluation process and present the result in a format that is both accurate and easily understandable. The use of standard form also helps in maintaining precision and avoiding ambiguity when dealing with numbers that have many digits or decimal places.
Conclusion
In this article, we have demonstrated the process of using log tables to evaluate a complex expression and express the answer in standard form. The expression we evaluated was:
√(853 * 4) / (54 * 81)
We began by applying logarithms to the expression, which allowed us to transform the complex operations of multiplication, division, and square roots into simpler addition, subtraction, and multiplication. This step is crucial for leveraging the properties of logarithms and making the calculation more manageable. Next, we used log tables to find the logarithmic values of the individual numbers in the expression. Log tables provide the mantissa of the logarithm, and we determined the characteristic based on the standard form of the numbers. By accurately finding these logarithmic values, we laid the foundation for the subsequent calculations. After obtaining the logarithmic values, we substituted them back into the equation and simplified. This involved performing arithmetic operations such as addition, subtraction, and multiplication with the logarithmic values. This step reduced the expression to a single logarithmic value, which represented the logarithm of our final answer. To find the actual value, we needed to find the antilogarithm. We used the antilogarithm table to find the number corresponding to the mantissa and adjusted the decimal point based on the characteristic. This step converted the logarithmic value back into the original numerical value. Finally, we expressed the answer in standard form. Standard form is a way of writing numbers as a product of a number between 1 and 10 and a power of 10. This notation is particularly useful for representing very large or very small numbers concisely. By expressing the final answer in standard form, we provided a clear and easily understandable representation of the result. The final answer, expressed in standard form, was:
1. 335 * 10⁻²
The process of using log tables and standard form is a valuable skill in mathematics and science. It allows us to handle complex calculations involving multiplication, division, and roots efficiently and accurately. While calculators have become ubiquitous, understanding how to use log tables provides a deeper understanding of logarithms and their properties. Moreover, it is a useful skill in situations where calculators are not available. By mastering the steps outlined in this article, one can confidently evaluate complex expressions and express the results in a clear and concise manner. The combination of logarithmic properties, log tables, and standard form provides a powerful toolkit for solving a wide range of mathematical and scientific problems. This methodical approach not only ensures accuracy but also enhances problem-solving skills and mathematical intuition. The ability to break down complex problems into simpler steps and apply appropriate techniques is a hallmark of mathematical proficiency, and the use of log tables and standard form exemplifies this approach.