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Chapter 13: Problem 5

Evaluate each logarithm. Do not use a calculator. $$\log \frac{1}{1000}$$

Short Answer

Expert verified
The logarithm \(\log{\frac{1}{1000}}\) can be evaluated as follows: Rewrite the fraction as a power of 10, \(\frac{1}{1000} = 10^{-3}\). Then, using the fact that \(\log_{10}(10^x) = x\), we find that \(\log{10^{-3}} = -3\). Therefore, the final answer is \(\log \frac{1}{1000} = -3\).

Step by step solution

01

Identify the base and the argument

In the given logarithm \(\log{\frac{1}{1000}}\), the base is 10 (since it is a common logarithm) and the argument is \(\frac{1}{1000}\).
02

Rewrite the fraction as a power of 10

We need to rewrite the fraction \(\frac{1}{1000}\) as a power of \(10^x\), where x is an integer. Notice that we can rewrite the fraction as follows: \(\frac{1}{1000} = \frac{1}{10^3} = 10^{-3}\).
03

Evaluate the logarithm

Now, we can use the fact that \(\log_{10}(10^x) = x\) to evaluate the given logarithm. In this case, x is -3, so we have: \(\log{10^{-3}} = -3\). So, the final answer is: \[ \log \frac{1}{1000} = -3 \]

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Key Concepts

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

Base of Logarithms
When we talk about logarithms, the "base" is a fundamental concept. The base of a logarithm tells us what number we are repeatedly multiplying by to get to another number. In the logarithmic expression \(\log_b{(x)}\), \(b\) is the base, and \(x\) is the number we want to reach. This reads as "log base \(b\) of \(x\)."

For example, in \(\log_{10}(100)\), the base is 10. We are asking what power of 10 gives us 100. The answer is 2, because \(10^2 = 100\).

The base is crucial because it determines the scale of the math involved.
  • A base of 10, known as the common logarithm, is widely used in science and engineering.
  • Different bases are used in various applications, such as base 2 for binary systems in computer science.
Powers of Ten
Understanding powers of ten can help you easily evaluate logarithms involving fractions. Powers of ten are expressions like \(10^1 = 10\), \(10^2 = 100\), and so forth. Negative powers, such as \(10^{-1} = 0.1\), are used for fractions

To express a fraction using a power of ten, you convert the denominator into a power of ten and write it as an exponent. For example, \(\frac{1}{1000}\) can be rewritten as \(10^{-3}\) because \(1000 = 10^3\). Therefore, \(\frac{1}{1000} = 10^{-3}\).

This idea is handy when dealing with logarithmic problems, as it simplifies calculations and allows you to focus on the exponents.
Evaluating Logarithms
Logarithms may seem tricky, but they simplify many mathematical processes. When evaluating a logarithm, you're essentially finding the power needed for the base to achieve a certain number.

To evaluate \(\log_{10}\left(\frac{1}{1000}\right)\):
  • First, rewrite \(\frac{1}{1000}\) as a power of ten, which is \(10^{-3}\).
  • Use the property of logarithms: \(\log_{10}(10^x) = x\).
  • The exponent (-3) is the value of the logarithm, so \(\log_{10}(10^{-3}) = -3\).
This simple method allows you to evaluate logarithms without a calculator, just by understanding powers of ten and the nature of logarithms.
Common Logarithm
Common logarithms are logarithms with base 10, denoted as \(\log(x)\), which is shorthand for \(\log_{10}(x)\). They're called "common" because they frequently appear in scientific and engineering calculations where the scale is base 10.

With common logarithms, the problem \(\log \left(\frac{1}{1000}\right)\) already assumes a base of 10. This format is convenient for human-centered scales, such as:
  • Scales used in decibels, which measure sound intensity.
  • The pH scale, measuring acidity or alkalinity.
Recognizing a common logarithm lets you bypass stating the base, easing both writing and calculations. Understanding this convention is key to navigating many practical logarithmic applications.

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