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Chapter 10: Problem 15

Evaluate each logarithm to four decimal places. $$ \log 0.0326 $$

Short Answer

Expert verified
-1.4860

Step by step solution

01

Identify the base

Assume the base of the logarithm is 10 as no base is specified. This means \( \log(x) \) typically refers to \( \log_{10}(x) \).
02

Calculate the logarithm

Use a calculator to compute \( \log_{10}(0.0326) \). Enter 0.0326 and press the log function button to get the value.
03

Round the result

Round the obtained value to four decimal places. The value obtained from the calculator is approximately -1.4860.

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Key Concepts

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

logarithm base 10
In mathematics, logarithms help us understand and work with large numbers more easily. The base of a logarithm is crucial for its calculation. When you see a logarithm without a specified base, it's typically base 10, also known as the common logarithm. Base 10 logarithms are written as \(\log(x)\) and are often used in scientific calculations and various applications. For example, \(\log_{10}(1) = 0\) because \(10^0 = 1\), and \(\log_{10}(10) = 1\) because \(10^1 = 10\). Understanding base 10 is essential for evaluating logarithms correctly.
calculator usage
Using a calculator to find logarithms is a straightforward process. Here's how you can do it:
  • Ensure your calculator is in normal mode (not scientific or graphing mode)
  • Enter the number for which you want to find the logarithm, in this case, 0.0326
  • Press the 'log' button on your calculator (usually labeled as 'log')
  • Read the displayed value
For our example, entering 0.0326 and pressing 'log' will give you approximately -1.4860. Calculators simplify logarithmic calculations tremendously and ensure accuracy.
rounding numbers
Rounding numbers makes our calculations and results more manageable. Often, we round numbers to a specified number of decimal places. Here's how to round to four decimal places accurately:
  • Locate the digit in the fourth decimal place
  • Look at the digit immediately to its right (the fifth decimal place)
  • If the fifth digit is 5 or higher, increase the fourth digit by one
  • If the fifth digit is less than 5, leave the fourth digit as it is
For instance, with the logarithm value -1.486015, we look at the fifth decimal place, which is 1. Since 1 is less than 5, we do not change the fourth decimal place, resulting in -1.4860.

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{0.04 x}=\sqrt{3} $$

Determine whether common logarithms or natural logarithms would be a better choice to use for solving each equation. Do not actually solve. $$ 10^{3 x+1}=13 $$

Based on selected figures obtained during the years \(1970-2015,\) the total number of bachelor's degrees earned in the United States can be modeled by the function $$ D(x)=792,377 e^{0.01798 x} $$ where \(x=0\) corresponds to \(1970, x=5\) corresponds to \(1975,\) and so on. Approximate, to the nearest unit, the number of bachelor's degrees earned in 2015. (Data from U.S. National Center for Education Statistics.)

Solve each equation. Give exact solutions. $$ \log _{4}(x-3)^{3}=4 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} \frac{9}{2} $$
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