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Chapter 12: Problem 28

Simplify. $$ \log _{10} 0.1 $$

Short Answer

Expert verified
\( \log_{10} 0.1 = -1 \)

Step by step solution

01

Understand the logarithm function

Logarithms are the inverse process of exponentiation. The logarithm \log_{10} x represents the exponent to which 10 must be raised to produce the number \( x \). So, if \( y = \log_{10} 0.1 \), then \( 10^y = 0.1 \).
02

Rewrite 0.1 as a power of 10

Note that 0.1 can be expressed as a power of 10: \( 0.1 = 10^{-1} \). This means \( 0.1 = 10^{-1} \).
03

Apply the logarithm rule

Using the property of logarithms that \( \log_b (b^x) = x \), we find the logarithm of a power of 10. Therefore, \( \log_{10} (10^{-1}) = -1 \).

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Key Concepts

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

inverse of exponentiation
To understand logarithms, we need to first grasp the idea of inverse operations. Exponentiation is the process of raising a number to a power. For example, in the expression 10^2 = 100, 10 is raised to the power of 2 to yield 100. The inverse of this operation is a logarithm. A logarithm asks the question: 'To what power must the base be raised, to obtain this number?' So, \( \log_{10} 100 = 2 \) because 10 raised to the power of 2 equals 100. Remember, logarithms and exponentiation are opposite operations. Whenever you see a logarithm, think of it as the reverse of raising a number to a power.
logarithm properties
Logarithms have several key properties that make them easier to work with. Here are some essential ones:
  • Product Property: \( \log_b (xy) = \log_b x + \log_b y \). This means the logarithm of a product is the sum of the logarithms.
  • Quotient Property: \( \log_b (x/y) = \log_b x - \log_b y \). The logarithm of a fraction is the difference of the logarithms.
  • Power Property: \( \log_b (x^n) = n \log_b x \). The logarithm of a power is the exponent times the logarithm of the base.
  • Change of Base Formula: \( \log_b x = \frac{\log_k x}{\log_k b} \). This allows you to change the base of the logarithm.
Using these properties can make it much easier to solve logarithm problems.
power of ten
The power of ten is a fundamental concept in logarithms, especially when dealing with common logarithms (base 10). When we say a number is a power of ten, we mean it can be written as 10 raised to an exponent. For example, \textbf{10^3 = 1000}, where 1000 is the result of 10 raised to the 3rd power.

In our example exercise, we have 0.1 . This can be rewritten as a power of ten: \( 0.1 = 10^{-1} \). On the logarithmic scale, the log base 10 of 0.1 directly tells us the exponent: \( \ \log_{10} 10^{-1} = -1 \). Recognizing numbers as powers of ten helps us quickly simplify and solve logarithmic expressions.

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