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Chapter 12: Problem 10

Find each logarithm. Give approximations to four decimal places. \(\log 457.2\)

Short Answer

Expert verified
2.6608

Step by step solution

01

- Understand the Problem

The task requires finding the logarithm of 457.2 to the base 10. This means that we need to compute \(\log_{10} 457.2\) and provide the result to four decimal places.
02

- Use a Calculator

Enter 457.2 into a scientific calculator and press the \('\log'\) button. The calculator should display the logarithm value to several decimal places.
03

- Approximating the Logarithm Value

The calculator will provide a number, and it should be approximated to four decimal places for precision. Read the value from the calculator鈥檚 display.
04

- Round to Four Decimal Places

If the calculator shows \(\log 457.2\) as 2.660775, round this number to four decimal places. The result is 2.6608.

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Key Concepts

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

logarithmic functions
Logarithmic functions are an essential topic in mathematics that often appears in various fields such as science, engineering, and finance. A logarithm answers the question: 'To what exponent must we raise a certain base to obtain a specific number?' For example, in the expression \(\text{log}_{10} 100\), we are looking for the exponent to which 10 must be raised to get 100. The answer is 2 because 10 raised to the power of 2 is 100. The notation takes the form \[ \text{log}_b (x) = y \] This means that b raised to the power of y equals x. When the base is 10, it鈥檚 called a common logarithm, often written simply as \(\text{log}\) without the base.
scientific calculator use
Using a scientific calculator is key to solving logarithms, especially for non-integer or large values like 457.2. Here鈥檚 a simple guide to find \(\text{log}\) values:
  • Turn on the scientific calculator.
  • Enter the number for which you want to calculate the logarithm (like 457.2).
  • Press the \('\text{log}'\) button.
The calculator will display the result immediately. If the display shows more than four decimal places, you should round it to the specified precision. Practice using the calculator to familiarize yourself with this process.
decimal approximation
Decimal approximation is rounding a number to a specific number of decimal places to make it easier to read and work with. Here鈥檚 how you can approximate to four decimal places:
  • Identify the fifth decimal place. If this digit is 5 or more, round the fourth place up.
  • If the fifth decimal place is less than 5, keep the fourth decimal place as it is.
For example, if a calculator shows \(\text{log} 457.2 = 2.660775\), you look at the fifth decimal place (7). Because 7 is more than 5, you round the fourth decimal up. Therefore, 2.660775 rounded to four decimal places is 2.6608. Practicing this rounding technique ensures accuracy in your approximations.

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Most popular questions from this chapter

If \(f(x)=4^{x},\) find each value indicated. In Exercise 50, use a calculator, and give the answer to the nearest hundredth. \(f\left(\frac{1}{2}\right)\)

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=\sqrt[3]{x+2}\)

Find each logarithm. Give approximations to four decimal places. \(\log 98\)

Which point lies on the graph of \(f(x)=3^{x} ?\) A. \((1,0)\) B. \((3,1)\) C. \((0,1)\) D. \(\left(\sqrt{3}, \frac{1}{3}\right)\)

Find each logarithm. Give approximations to four decimal places. \(\log 0.1741\)
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