Problem 71 - Algebra 2 and Trigonometry

Question

If \(\mathrm{g}(x)=\log _{10} x,\) find \(\mathrm{g}(0.001)\)

Step-by-Step Solution

Answer

\( g(0.001) = -3 \).

Step 1: Understand the Function

The given function is \( g(x) = \log_{10} x \). This is the logarithm function with base 10, often referred to as the common logarithm.

Step 2: Substitute the Given Value

We need to find \( g(0.001) \). Substitute \( x = 0.001 \) into the function: \( g(0.001) = \log_{10} 0.001 \).

Step 3: Express 0.001 as a Power of 10

Notice that \( 0.001 = 10^{-3} \). The number 0.001 can be rewritten as a power of 10 for easier calculation.

Step 4: Apply the Logarithmic Identity

Using the identity \( \log_{10} (a^b) = b \cdot \log_{10} (a) \), where \( a = 10 \) and \( b = -3 \), we have \( \log_{10} 10^{-3} = -3 \cdot \log_{10} 10 \).

Step 5: Evaluate the Common Logarithm

Since \( \log_{10} 10 = 1 \), this simplifies to \( g(0.001) = -3 \cdot 1 = -3 \).

Key Concepts

Base 10 Logarithms

In everyday mathematics, base 10 logarithms are quite common and are known as "common logarithms." The base 10 logarithm of a number is essentially the power that 10 must be raised to get that number. For example, the base 10 logarithm of 100 is 2, because 10 raised to the power of 2 gives you 100.
Understanding common logarithms is crucial as they are often used in scientific calculations, especially when dealing with data and measurements across diverse scales. When you see the term "log" without an explicitly defined base, it is usually assumed to be base 10.

• Example: If someone writes \( \log 1000 \), the implied base is 10, and the solution is 3, because \( 10^3 = 1000 \).
• Notation: We often write base 10 logarithms as either \( \log_{10} x \) or simply \( \log x \).

Evaluating Logarithms

Evaluating logarithms involves finding the exponent that the base must be raised to, in order to get the number in question. Let's explore an example to make this clearer. Consider the function \( g(x) = \log_{10} x \). To evaluate \( g(0.001) \), we perform the following steps:
1. **Substitute the Value**: Place 0.001 into the logarithm function: \( g(0.001) = \log_{10} 0.001 \).
2. **Express in Powers of 10**: Express 0.001 as a power of 10. In this case, \( 0.001 = 10^{-3} \).
3. **Apply Logarithm Rule**: Use the identity \( \log_{10} (a^b) = b \cdot \log_{10} a \). Therefore, \( \log_{10} 10^{-3} = -3 \cdot \log_{10} 10 \).
4. **Complete the Evaluation**: Since \( \log_{10} 10 = 1 \), this simplifies to \( -3 \times 1 = -3 \).
5. **Conclusion**: So, \( g(0.001) = -3 \), meaning that 10 raised to the power of -3 equals 0.001.

Logarithm Properties

Logarithms have some handy properties that make calculations and transformations much easier. These rules help simplify complex expressions and convert them into more manageable forms.
Here are three key properties to remember:

• **Product Rule**: \( \log_b (mn) = \log_b m + \log_b n \). This tells us that the log of a product is the sum of the logs.
• **Quotient Rule**: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). This rule simplifies division inside the log.
• **Power Rule**: \( \log_b (m^n) = n \cdot \log_b m \). This makes handling powers inside a log straightforward and is used directly in evaluating expressions like \( \log_{10} 10^{-3} \).

These properties are essential for simplifying and solving problems involving logarithms. They enable you to break down complex logarithmic expressions into more basic parts, which can then be easily calculated.