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Chapter 9: Problem 29

Simplify. $$ \log _{10} 0.01 $$

Short Answer

Expert verified
The simplified form is \(-2\).

Step by step solution

01

- Understand the logarithmic form

Recall that a logarithm represents the exponent to which a base number must be raised to yield a given number. Here, we need to determine the value of \(\log_{10} 0.01\).
02

- Rewrite the number in exponential form

Express \(0.01\) as a power of 10. \(0.01\) can be written as \(10^{-2}\). So, \(\log_{10} 0.01\) is equivalent to \(\log_{10} 10^{-2}\).
03

- Apply the logarithm power rule

Use the logarithm power rule, \(\log_{b}(a^c) = c\log_{b}(a)\). Applying this rule to \(\log_{10} 10^{-2}\), we get \(\log_{10} 10^{-2} = -2 \cdot \log_{10} 10\).
04

- Simplify the logarithm

Since \(\log_{10} 10 = 1\), we can simplify the expression \( -2 \cdot \log_{10} 10 \) to \(-2 \cdot 1 = -2\).

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Key Concepts

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

logarithmic form
To understand logarithms better, let's dive into the concept of 'logarithmic form'. Think of a logarithm as an exponent. It tells us the power to which we must raise a base to get a certain number. The general form is \(\text{log}_b(a) = c\). Here, \(b\) is the base, \(a\) is the number, and \(c\) is the exponent. For example, in our exercise, we have \(\text{log}_{10}(0.01)\). This asks, 'To what power must 10 be raised to get 0.01?'
exponential form
Converting numbers to exponential form makes logarithms easier to handle. Exponential form is just a way to express numbers as powers of a base. For instance, \(0.01\) can be written as \(10^{-2}\). In exponential form, we state that \(10^{-2} = 0.01\). Therefore, our logarithmic expression \( \text{log}_{10}(0.01) \) becomes \( \text{log}_{10}(10^{-2}) \).
logarithm power rule
The logarithm power rule simplifies expressions where the argument is a power of the base. The rule states \( \text{log}_b(a^c) = c \text{log}_b(a) \). To apply this rule to our problem, \( \text{log}_{10}(10^{-2})\), we use \( \text{log}_{10}(10^{-2}) = -2 \text{log}_{10}(10) \). This reduces our expression by taking the exponent \(-2\) and multiplying it by \( \text{log}_{10}(10) \).
simplifying logarithms
Now, let's finish the process by simplifying the logarithm completely. Given that \( \text{log}_{10}(10) = 1 \), our expression \( -2 \text{log}_{10}(10) \) simplifies further. So, \( -2 \text{log}_{10}(10) = -2 \times 1 = -2 \). Therefore, \(\text{log}_{10}(0.01) = -2\).

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