Problem 48 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\log (3 x+7)-\log x\)

Step-by-Step Solution

Answer

The simplified single logarithmic expression is \(\log(3 + 7/x)\).

Step 1: Identify the Type of Logarithmic Property

Looking at the expression \(\log (3 x+7)-\log x\), it can be seen that it is subtraction of two log functions. This comes directly under the log property known as quotient property which states \(\log(a) - \log(b) = \log(a/b)\). This property can be used to simplify the given expression.

Step 2: Apply the Quotient Rule

According to the quotient rule, the given expression can be rewritten as : \(\log((3x + 7) / x)\). So replace \(\log (3 x+7)-\log x\) with \(\log((3x+7) / x)\) as per the quotient rule.

Step 3: Simplify the Expression

The expression inside the logarithm ((3x+7) / x) simplifies to 3 + 7/x. So, the entire expression simplifies to \(\log(3 + 7/x)\). Always re-check your simplification to make sure it is accurate.

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics that allow us to deal with exponential relationships by transforming multiplication and division into addition and subtraction. The properties of logarithms are key to simplifying these expressions. Here are the main properties that are widely used:

• Product Property: \( \log(a) + \log(b) = \log(ab) \)
• Quotient Property: \( \log(a) - \log(b) = \log(a/b) \)
• Power Property: \( \log(a^b) = b \cdot \log(a) \)

By using these properties, logarithmic expressions can be manipulated and simplified into more manageable forms. This process of changing and combining logarithms is often referred to as "condensing." Understanding these properties forms the foundation for working with logarithmic expressions effectively.

Quotient Property

The quotient property of logarithms is a crucial tool when simplifying expressions involving the subtraction of two logs. This property states that if you have two logarithms with the same base that are subtracted, it can be rewritten as the logarithm of a division.
For example, given \[ \log(a) - \log(b) = \log \left( \frac{a}{b} \right) \]This allows us to turn the subtraction into a single logarithm expression involving division.
In our exercise, we see this applied directly:

• The original expression: \( \log(3x + 7) - \log(x) \)
• Simplified using the quotient property: \( \log \left( \frac{3x + 7}{x} \right) \)

This property is immensely useful for not only simplifying expressions but also for solving logarithmic equations by transforming them into simpler forms.

Condensing Logarithms

Condensing logarithms refers to the process of combining several logarithmic terms into a single term. This is especially useful in simplifying expressions and solving equations. By using the properties of logarithms, including the quotient and product properties, complex logarithmic expressions are reduced to simpler, single-logarithm forms.
In our specific problem, the condensing process involved:

• Starting with \( \log(3x + 7) - \log(x) \)
• Applying the quotient property to write as \( \log \left( \frac{3x + 7}{x} \right) \)
• Simplifying further to \( \log \left( 3 + \frac{7}{x} \right) \)

Condensing is highly valuable when evaluating or further processing logarithmic expressions, as it allows equations to be more easily analyzed and understood.