Problem 567 - Precalculus

Question

Rewrite \(\log (17 a \cdot 2 b)\) as a sum.

Step-by-Step Solution

Answer

\(\log (17) + \log (a) + \log (2) + \log (b)\)

Step 1: Understand Logarithmic Properties

To rewrite a logarithmic expression, you need to know some of the basic properties of logarithms. One important property is that the logarithm of a product can be written as the sum of the logarithms of the factors: \[\log(xy) = \log x + \log y.\] This property will be useful to rewrite the given expression as a sum.

Step 2: Break Down the Expression

First, consider the expression inside the logarithm, which is \(17a \cdot 2b\). Recognize that this expression is a product of three factors: \(17\), \(a\), and \(2b\). Apply the product property for logarithms to break down this product.

Step 3: Apply the Logarithm Product Rule

Use the property from Step 1: \[\log (17a \cdot 2b) = \log (17) + \log (a) + \log (2b).\]Now, the expression is separated into a sum of logarithms. But notice that we still have a product \(2b\) which we can further break down as well.

Step 4: Further Break Down \(\log(2b)\)

Using the same logarithm property, apply the rule to \(\log (2b)\): \[\log (2b) = \log (2) + \log (b).\] Substitute this back into the expression to fully simplify it as a sum of individual logarithms.

Step 5: Write the Final Expression

Substitute the result from Step 4 back into the expression: \[\log (17) + \log (a) + \log (2b) = \log (17) + \log (a) + \log (2) + \log (b).\]Thus, the original expression \(\log (17a \cdot 2b)\) is rewritten as the sum \(\log (17) + \log (a) + \log (2) + \log (b)\).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a handy tool when working with logarithmic expressions. It makes simplifying complex expressions much easier. This rule states that when you are taking the logarithm of a product, you can separate it into a sum of the logarithms of the individual factors. Here's what this looks like:

• If you have \(\log(xy)\), you can rewrite it as \(\log x + \log y\).
• This is an essential property because it allows you to break down, and thus simplifies, expressions.
• Using this rule makes it easier to work with and interpret logarithmic expressions in equations.

You can think of it like splitting a complicated job into smaller tasks, making each part easier to handle.
By understanding and using the product rule, you can make sense of what might initially seem like complex mathematical puzzles.

Logarithm of a Product

When dealing with a logarithm of a product, it's essential first to identify the components involved in that product. In the expression \(\log (17a \cdot 2b)\), there are different factors involved:

• Recognize you are dealing with the factors: \(17\), \(a\), \(2\), and \(b\).
• Each of these factors can be separated, thanks to the logarithmic properties, into individual logarithms.
• This "separation" is key to simplifying or rewriting the expression.

By applying the product rule, you see how the product's logarithm can be rewritten as a sum:
\(\log (17a \cdot 2b)\) becomes \(\log (17) + \log (a) + \log (2b)\). This step-by-step breakdown helps make the process clear and logical.
You just distribute the log across each factor of the product, which is a straightforward application of the product rule.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using logarithmic properties to break down expressions into simpler forms. This task is all about making the expression easier to work with or understand. For example:

• Once you've identified all the individual components provided by the logarithm of a product, you can simplify these into distinct parts.
• Take \(\log(2b)\), for instance. It can further be broken down into \(\log(2) + \log(b)\).
• This results in a totally simplified form of the expression, allowing you to focus on each piece separately.

The final expression \(\log (17) + \log (a) + \log (2) + \log (b)\) elucidates how each factor contributes its part to the total sum.
The process of simplification is instrumental in mathematical analysis and problem-solving. It reduces complexity, making it easier to communicate and understand various mathematical concepts.
By mastering this skill, you'll find many logarithmic issues far less daunting.