Problem 38 - Elementary and Intermediate Algebra

Question

Write logarithm as a sum. Then simplify, if possible. \(\log 10 a b c\)

Step-by-Step Solution

Answer

\( \log 10 a b c = 1 + \log a + \log b + \log c \)

Step 1: Identify the Expression

We need to express \( \log 10abc \) as a sum of separate logarithms. Notice that we have a product inside the logarithm: \( 10 \times a \times b \times c \).

Step 2: Apply Logarithm of a Product Rule

Use the rule \( \log(xy) = \log x + \log y \). Apply it to \( \log(10 \times a \times b \times c) \). We can expand it as \( \log 10 + \log a + \log b + \log c \).

Step 3: Simplify Logarithms

Notice that \( \log 10 \) simplifies to 1, since 10 is the base of the common logarithm. Replace \( \log 10 \) with 1 to simplify: \( 1 + \log a + \log b + \log c \).

Key Concepts

Logarithm Properties

Logarithms are mathematical expressions that help us understand and work with exponential relationships, using the concept of the inverse of raising a number to a power. When working with logarithms, there are some important properties that allow us to manipulate these expressions more easily:

• The Product Rule: This rule lets us break down the logarithm of a product into a sum of logarithms. It states that \( \log(xy) = \log x + \log y \).
• The Quotient Rule: This rule is similar but deals with division. It states that \( \log(x/y) = \log x - \log y \).
• The Power Rule: This allows us to take the exponent out of the logarithm's argument: \( \log(x^n) = n \cdot \log x \).

Understanding these properties is essential for simplifying logarithmic expressions. By applying them, you can transform complicated expressions into more manageable ones.

Product Rule in Logarithms

The product rule in logarithms is a valuable tool that makes dealing with multiple factors straightforward. As mentioned, the product rule states:

• \( \log(xy) = \log x + \log y \)

This simplifies the process of dealing with logarithmic expressions that contain products. For example, if you need to find \( \log(abc) \), you can decompose it into three separate logarithms:

• \( \log(abc) = \log a + \log b + \log c \)

Each logarithm can be calculated or simplified separately, giving you the flexibility to work with individual components rather than tackling the complex product all at once. By using this rule, you can rewrite expressions such as \( \log 10abc \) into a sum like \( \log 10 + \log a + \log b + \log c \), making it much easier to simplify later.

Simplifying Logarithms

Simplifying logarithms typically involves rewriting a logarithmic expression in its simplest form. After applying the product rule, you often end up with a series of logarithms you can evaluate more straightforwardly.A common simplification involves recognizing when a logarithm can be directly calculated. For instance, when the base of the logarithm matches the number inside it, the result is simply 1:

• For common (base 10) logarithms, \( \log 10 = 1 \).
• For natural logarithms (base \( e \)), \( \ln e = 1 \).

In our example \( \log 10abc \), after expressing it as \( \log 10 + \log a + \log b + \log c \), we simplify \( \log 10 \) to 1. This transforms our expression into \( 1 + \log a + \log b + \log c \).Simplifying logarithmic expressions is crucial in many mathematical contexts as it can help make sense of the calculations and yield results more efficiently.