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Chapter 7: Problem 20

Demonstrate numerically the propertics of logarithms. \(\log \frac{1}{1000}=-\log 1000\)

Short Answer

Expert verified
\(\log(\frac{1}{1000}) = -3\), which equals \(-\log(1000)\) as demonstrated by logarithm properties.

Step by step solution

01

Understanding Logarithm Properties

The logarithm of a quotient is equal to the difference of the logarithms. Formally, if we have two positive numbers, a and b, then \(\log(\frac{a}{b}) = \log(a) - \log(b)\). This means that \(\log(\frac{1}{1000})\) is equal to \(\log(1) - \log(1000)\).
02

Evaluate Logarithms Individually

We have \(\log(1)\) which is always 0 because any number to the power of 0 is 1. Also, \(\log(1000)\) is 3 since 10 to the power of 3 is 1000. So, we substitute these values into the equation: \(0 - 3\).
03

Simplify the Expression

Subtract the values to get the result for \(\log(\frac{1}{1000})\). The calculation is \(0 - 3 = -3\).
04

Recognize Negative Logarithms

Notice that the negative log of a number (in this case 1000) is equal to the log of its reciprocal, which means that \(-\log(1000) = \log(\frac{1}{1000})\).

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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 the inverse of exponential functions, and they play a crucial role in various branches of mathematics and applied sciences. At their core, they answer the question:

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Most popular questions from this chapter

Demonstrate numerically the propertics of logarithms. \(\ln 5^{3}=3 \ln 5\)

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