Problem 41 - Algebra 2

Question

Evaluate each expression. \(\log _{10} 0.001\)

Step-by-Step Solution

Answer

\(\log_{10} (0.001) = -3\)

Step 1: Understand the Logarithm Function

The logarithm function \(\log_b(x)\) answers the question: "What power do we need to raise \(b\) to, in order to get \(x\)?" Here, \(b\) is the base and \(x\) is the number for which we're finding the logarithm.

Step 2: Write the Expression in Logarithmic Terms

In this problem, we have the expression \(\log_{10}(0.001)\). This asks: "What power must 10 be raised to in order to result in 0.001?"

Step 3: Convert to Exponential Form

To find \(\log_{10}(0.001)\), write the number 0.001 as a power of 10. We have 0.001 = \(10^{-3}\). Now, since the base is 10 and we are asked for \(\log_{10}(0.001)\), identify that \(10^{-3}\) corresponds directly to our answer.

Step 4: Identify the Logarithm Value

Since 0.001 is \(10^{-3}\), it follows that \(\log_{10}(0.001) = -3\). This is because 10 raised to the power of -3 equals 0.001.

Key Concepts

Base 10 Logarithms

The base 10 logarithm, often simply called the "common logarithm", is an important and widely used function in mathematics. When you see "log" without a base indicated, it's usually assumed to be base 10. This type of logarithm answers the question: "To what power must 10 be raised to obtain a certain number?" For example, to evaluate \(\log_{10}(0.001)\), we want to know which power, when applied to 10, results in 0.001.

• In this specific case, the base is 10.
• The logarithm asks for the exponent that will yield the number 0.001.
• Commonly, calculators have a direct button labeled "log" representing \(\log_{10}\).

Understanding base 10 logarithms is fundamental, especially because they appear in various scientific and everyday contexts, such as the Richter scale for earthquakes or the pH scale in chemistry. They help us simplify and manage large quantities exponentially, playing a critical role in data analysis and scientific computation.

Properties of Logarithms

Logarithms have a set of key properties that make them useful for simplifying complex calculations. These properties are helpful when evaluating logarithms or manipulating expressions involving logs.

• Product Property: \(\log_b(MN) = \log_b(M) + \log_b(N)\) - Logarithm of a product is the sum of the logarithms.
• Quotient Property: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\) - Logarithm of a quotient is the difference of the logarithms.
• Power Property: \(\log_b(M^n) = n \cdot \log_b(M)\) - Logarithm of an exponentiated number is the exponent times the logarithm of the base.
• Base Change Property: Allows conversion between different logarithm bases using \(\log_b(M) = \frac{\log_k(M)}{\log_k(b)}\).

These properties make logarithms adaptable and they transform multiplicative relationships into additive ones, which are easier to manipulate. Therefore, these rules not only aid in solving the given exercise more intuitively but also widen our toolkit for dealing with exponential growth scenarios or scaling problems.

Exponential Form

Converting a logarithmic equation into its exponential form can simplify understanding and solving it. Essentially, the exponential form is the reverse of a logarithmic function. The statement \(\log_b(x) = y\) can be translated into the exponential form of \(b^y = x\). This transformation is highly useful for solving logarithmic problems because considering numbers in terms of base and exponent can make hidden patterns more visible.

• For our case, \(\log_{10}(0.001) = -3\) translates to \(10^{-3} = 0.001\).
• Rewriting logarithms as exponents can also aid in recognizing binary relationships that aren't instantly obvious.
• In exponential form, understanding the root or fractional exponents can become straightforward.

The insight provided by rewriting logarithmic expressions as exponential equations reveals their relationship to real-world scenarios, such as calculating interest rates or understanding radioactive decay, where exponential patterns frequently occur.