Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the common logarithm of each number to the nearest hundredth. $$ 0.1 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Common Logarithms

The common logarithm of a number is the logarithm to base 10. It is often denoted as \( \log_{10} \) or simply \( \log \). In this exercise, we need to find \( \log_{10}(0.1) \).

Step 2: Using the Logarithmic Identity

Recall the logarithmic identity: \( \log_{10}(10^n) = n \). Our task is to express 0.1 as a power of 10. Note that 0.1 is equivalent to \( 10^{-1} \).

Step 3: Applying the Identity

Using the identity from the previous step, we find that \( \log_{10}(10^{-1}) = -1 \). Thus, \( \log_{10}(0.1) = -1 \).

Step 4: Rounding to the Nearest Hundredth

No rounding is actually necessary here, as \(-1\) is an exact value and already has two decimal places as \(-1.00\).

Key Concepts

Logarithmic Identity

When dealing with logarithms, a key concept is the logarithmic identity. This is a fundamental rule used to simplify logarithmic expressions. A logarithmic identity that you often encounter relates powers of ten. It states that for any real number \( n \), \( \log_{10}(10^n) = n \). This identity is incredibly useful because it allows you to compute the logarithm of numbers that can be expressed as a power of ten instantly. For example, if you know that a number can be rewritten as \( 10^{-1} \), you can use the logarithmic identity to find its common logarithm (base 10 logarithm). In the specific case of 0.1, recognizing it as \( 10^{-1} \) allowed us to apply this identity and directly arrive at the result \( \log_{10}(0.1) = -1 \).
This not only gives you a deeper understanding of how numbers relate to their logarithms but also provides an efficient way to solve logarithmic problems without using a calculator.

Base 10

Base 10, also known as the decimal system, is the numeral system most commonly used in daily life. It uses ten digits, from 0 to 9, to represent numbers. When we speak about logarithms in base 10, we refer to the common logarithm, which is widely used in science and engineering. One of the conveniences of base 10 is that it aligns perfectly with our usual numeric system, making it intuitive for calculations.
The common logarithm of a number \( x \) is denoted as \( \log_{10}(x) \) or simply \( \log(x) \). For instance, finding \( \log_{10}(0.1) \) involves expressing 0.1 as a power of 10, which is \( 10^{-1} \). Using the rules of exponents and the logarithmic identity, the result directly simplifies to -1. \( \log_{10} \) is particularly handy because it reduces complex multiplication and division operations into simpler addition and subtraction problems, thanks to the properties of logarithms.
In technological algorithms, common logarithms often turn multiplicative relationships into additive ones, streamlining calculations and making problem-solving more manageable.

Logarithmic Functions

Logarithmic functions are fascinating mathematical tools that solve the inverse problem of exponential functions. While exponential functions describe how quantities grow, logarithmic functions tell you how to revert this growth to identify the original amount. This makes understanding logarithmic functions incredibly useful both in theoretical and practical applications.

• They allow you to transform multiplicative relationships into additive, simplifying complex calculations.
• They are involved in various scientific calculations, such as pH levels in chemistry or decibel measurements in acoustics.
• They help unravel exponential growth processes, such as population growth, radioactive decay, and compound interest.

In the context of this exercise, the common logarithm is a specific type of logarithmic function where the base is 10. By identifying that 0.1 equals \( 10^{-1} \), the function \( \log_{10}(x) \) helps determine the power to which this base must be raised to get the number. This specific function type plays a crucial role in mathematics and science, simplifying phenomena that involve exponential changes.