Problem 45 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 4. $$ \log \frac{10 x}{y} $$

Step-by-Step Solution

Answer

\( 1 + \log x - \log y \)

Step 1: Apply the Quotient Rule

The logarithm of a fraction can be expressed as the difference of the logarithms of the numerator and the denominator. For the expression \( \log \frac{10x}{y} \), use the quotient rule: \[ \log \frac{a}{b} = \log a - \log b \]. This results in: \[ \log (10x) - \log y \].

Step 2: Apply the Product Rule

Now, consider \( \log (10x) \). The logarithm of a product can be expressed as the sum of the logarithms. For \( \log (10x) \), apply the product rule: \[ \log (ab) = \log a + \log b \]. This gives: \[ \log 10 + \log x \].

Step 3: Combine the Expressions

Substitute \( \log (10x) = \log 10 + \log x \) into the earlier expression from Step 1 so that the expression becomes: \[ \log 10 + \log x - \log y \].

Step 4: Simplify the Expression

Since \( \log 10 \) is a known value (equal to 1), substitute this into the expression: \[ 1 + \log x - \log y \]. This is the simplified form of the original logarithmic expression.

Key Concepts

Understanding the Quotient Rule

The quotient rule is an essential tool in working with logarithms. When you have the logarithm of a fraction, you can break it down into a simpler form using the difference of two logarithms. This helps make calculations easier to handle. Consider the expression \( \log \frac{10x}{y} \). Here, you are working with a fraction where 10x is the numerator and y is the denominator.

Using the quotient rule, we can rewrite this as:

• \( \log \frac{a}{b} = \log a - \log b \)

In our case, this expression changes to \( \log (10x) - \log y \). Breaking down fractions into individual logarithms using this rule is particularly helpful when dealing with complex expressions. It allows you to focus on simpler components, which can then be further simplified if possible.

Diving into the Product Rule

The product rule is another vital part of simplifying logarithmic expressions. It allows you to take the logarithm of a product and express it as a sum of logarithms. Working with sum terms is typically more convenient, especially when simplifying expressions.

For the term \( \log (10x) \) from the expression \( \log (10x) - \log y \), apply the product rule to separate it further:

• \( \log (ab) = \log a + \log b \)

Using this rule, \( \log (10x) \) becomes \( \log 10 + \log x \). This step transforms a single logarithm involving multiplication into several simpler terms. By doing so, each component, like \( \log 10 \), can be handled individually, making further simplification easier.

Mastering Simplification of Expressions

Simplification is the key to making complex expressions more manageable. Once you break down the problem using the quotient and product rules, the next step is to refine the expression. In our example, after applying these rules, we obtained:

\( \log 10 + \log x - \log y \).

This expression is almost complete, but there's one more simplification left. We know the value of \( \log 10 \) due to common logarithmic properties. Since \( \log 10 \) equals 1 when the base of the logarithm is 10, you can substitute it directly:

• \( 1 + \log x - \log y \)

By breaking down logarithmic expressions step by step, you can simplify challenging problems into more straightforward versions, allowing for easier analysis or computation. Remember, keeping expressions simple helps greatly with validation and understanding of the core concepts.