Problem 36 - Elementary and Intermediate Algebra

Question

Write logarithm as a sum. Then simplify, if possible. \(\log 1,000 r s\)

Step-by-Step Solution

Answer

\( \log (1000 r s) = 3 + \log(r) + \log(s) \)

Step 1: Apply the Product Property of Logarithms

The product property states that the logarithm of a product is the sum of the logarithms: \[ \log(a \, b) = \log(a) + \log(b) \]In this case, apply it to \(\log (1000 \, r \, s)\):\[ \log (1000 \, r \, s) = \log(1000) + \log(r) + \log(s) \]

Step 2: Evaluate the Logarithm of 1000

Recognize that \(1000 = 10^3\). Now, apply the power rule \(\log(b^e) = e \cdot \log(b)\):\[ \log(1000) = \log(10^3) = 3 \cdot \log(10) \]Since \(\log(10) = 1\), we have:\[ \log(1000) = 3 \]

Step 3: Write the Expression as a Sum

Substitute the evaluated logarithm from Step 2 into the expression from Step 1:\[ \log(1000) + \log(r) + \log(s) = 3 + \log(r) + \log(s) \]This is the expression written as a sum.

Key Concepts

Product Property of Logarithms

The product property of logarithms is a handy tool when you're dealing with the logarithm of a product. It helps you break down complex expressions into simpler parts. The property states that the logarithm of a product is the sum of the logarithms of each factor.
For example, if you have a logarithm like \( \log(a \, b) \), you can apply the product property as follows:

• \( \log(a \, b) = \log(a) + \log(b) \)

By using this property, you can turn the logarithm of a product into the sum of two separate logarithms. This makes calculations more manageable and is crucial for solving problems like \( \log(1000 \, r \, s) \) in the exercise. Here, the product property allows you to separate \( \log(1000) + \log(r) + \log(s) \), making it easier to evaluate each part individually.

Power Rule of Logarithms

The power rule of logarithms is another powerful tool that simplifies complex logarithmic expressions, especially when dealing with exponential numbers. The power rule states that the logarithm of a power can be rewritten by multiplying the exponent with the logarithm of the base.
Here's how it works: if you have a logarithm like \( \log(b^e) \), you can transform it as follows:

• \( \log(b^e) = e \cdot \log(b) \)

Using this property, you can simplify \( \log(1000) \). Recognize that \( 1000 = 10^3 \), which can then be rewritten under the power rule:

• \( \log(10^3) = 3 \cdot \log(10) \)

Since \( \log(10) = 1 \), you arrive at \( 3 \cdot 1 = 3 \).
By breaking it down this way, you turn a potentially difficult calculation into an easy one. This skill is very useful in mathematics, particularly when simplifying expressions like in the given exercise.

Simplifying Logarithms

Simplifying logarithms is about making expressions as concise and clear as possible. It often involves using logarithmic properties like the product property and the power rule to untangle and reduce expressions. In the given exercise, you encountered the expression \( \log(1000 \, r \, s) \).
Here's a simple route to simplify it:

• First, apply the product property: \( \log(1000) + \log(r) + \log(s) \).
• Then, evaluate \( \log(1000) \) using the power rule to find \( 3 \).
• Finally, combine it all into \( 3 + \log(r) + \log(s) \).

This step-by-step approach helps you arrive at a simplified version of the original family of logarithms. By following these steps, you can transform the complex into the straightforward, gaining clarity not just in your current problem but enhancing your overall understanding of logarithmic expressions.