Problem 51 - Precalculus with Calculus Previews

Question

In Problems \(51-56\), use the laws of logarithms in Theorem \(6.2 .1\) to rewrite the given expression as one logarithm. $$ \log _{10} 2+2 \log _{10} 5 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{10} 50 \).

Step 1: Identify the Logarithm Laws

In order to combine multiple logarithms into one, we will need to use the laws of logarithms. Specifically, we'll use the Product Rule and the Power Rule. The Product Rule states that \( \log_b (MN) = \log_b M + \log_b N \). The Power Rule states that \( \log_b (M^n) = n \log_b M \).

Step 2: Apply the Power Rule

First, apply the Power Rule to the term \( 2 \log_{10} 5 \). According to the rule, we can rewrite this term as \( \log_{10} (5^2) \) or \( \log_{10} 25 \).

Step 3: Apply the Product Rule

After applying the Power Rule, we have the expression \( \log_{10} 2 + \log_{10} 25 \). According to the Product Rule, this can be rewritten as a single logarithm: \( \log_{10} (2 \times 25) \).

Step 4: Simplify the Expression

Now, we calculate the product inside the logarithm: \( 2 \times 25 = 50 \). So, the expression becomes \( \log_{10} 50 \).

Key Concepts

Product Rule

The Product Rule is a fundamental aspect of logarithms, making the process of combining log expressions simpler. The Product Rule states that \( \log_b(MN) = \log_b M + \log_b N \). This rule shows how the logarithm of a product can be expressed as the sum of the logarithms of its factors, all sharing the same base. Let's take a closer look at what this means:

• If you have two numbers, \( M \) and \( N \), being multiplied inside a logarithm function, you can "break" this product into two separate logarithms that are added together.
• This rule is particularly useful for simplifying expressions that initially appear complex, by turning multiplication inside the log into addition outside it.
• In our original exercise, once we used the Power Rule, we needed to combine \( \log_{10} 2 \) and \( \log_{10} 25 \). Thanks to the Product Rule, this combined neatly into a single logarithm: \( \log_{10} 50 \), since \( 2 \times 25 = 50 \).

Remember, the Product Rule only applies when all logs are of the same base. It can be a powerful tool in making further simplifications, particularly helpful in solving logarithmic equations.

Power Rule

The Power Rule is another key concept in logarithms, essential for rewriting logarithmic expressions effectively. The rule is defined as \( \log_b(M^n) = n \log_b M \). By using this rule, we can transform logarithms with exponents, simplifying expressions in the process.

• This rule allows you to "pull down" an exponent in a logarithmic statement as a coefficient in front of the log.
• For instance, in our problem, the term \( 2 \log_{10} 5 \) involves multiplying the logarithm by 2. By using the Power Rule, we can rewrite this as \( \log_{10} (5^2) = \log_{10} 25 \).
• The Power Rule is practical, especially when dealing with equations where the exponential expression needs to be simplified for easier manipulation.

This approach helps to manage larger expressions by reducing and simplifying them, emphasizing the versatility of the Power Rule in many logarithmic transformations.

Simplifying Logarithms

Simplifying logarithms involves using rules such as the Product and Power Rules to condense expressions into a single logarithm. This makes complex logarithmic expressions much easier to work with, whether you are solving equations or just rewriting them for clarity. Here’s how simplification works:

• The first step is to identify components that can be combined according to known rules, ensuring all logarithms share the same base for consistency.
• To simplify \( \log _{10} 2 + 2 \log _{10} 5 \), we first applied the Power Rule, transforming the expression into \( \log_{10} 2 + \log_{10} 25 \).
• Next, using the Product Rule, these terms were combined into a single logarithm: \( \log_{10} (2 \times 25) \), which then simplifies to \( \log_{10} 50 \).

Through such a systematic approach, simplifying logarithms can turn multiple expressions into more manageable forms, aiding significantly in solving complex mathematical problems.