Problem 55 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$4 \ln (x+6)-3 \ln x$$

Step-by-Step Solution

Answer

The condensed logarithmic expression for the given expression is \( \ln \left( \frac{(x + 6)^4}{x^3} \right) \).

Step 1: Apply Logarithmic Properties

Here we apply the second logarithmic property, which allows a coefficient in front of the log to be moved as a power of the base inside the logarithm. We transform each logarithmic expression according to this rule: \(4 \ln (x+6)\) becomes \(\ln ((x+6)^4)\) and \(-3 \ln x\) becomes \(- \ln (x^3)\). The full expression now looks like \(\ln ((x+6)^4) - \ln (x^3)\).

Step 2: Combine Logarithmic Expressions

Now the first property of logarithms can be used to combine the two separate log expressions into one. This property allows subtraction to be transformed into division inside the logarithm: \( \ln a - \ln b = \ln(a/b)\). This gives us a single logarithmic expression: \(\ln \left( \frac{(x + 6)^4}{x^3} \right)\).

Step 3: Evaluate Logarithmic Expression

If the value of x is known, we can evaluate the logarithmic expression now. As no specific value for x was provided in the task, this is the final form of the expression.

Key Concepts

Properties of Logarithms

Logarithms have several important properties that can simplify complex expressions. Understanding these properties helps to work with logarithms more efficiently. One of the core properties is the ability to transform coefficients into exponents. This is known as the power rule of logarithms and is expressed as: \(a \log_b(c) = \log_b(c^a)\). This rule is extremely useful when condensing logarithmic expressions because it reduces multiplication to exponentiation, allowing for easier manipulation.
Another key property is the product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\). This rule allows you to break down or expand expressions that involve the multiplication of terms.
Finally, the quotient rule stated as \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\) enables the expression of division as a subtraction of logs. These properties collectively form the foundation for manipulating and condensing logarithmic expressions.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple log terms into a single logarithm, often through inverse operations of the logarithmic properties. This process makes equations easier to solve or evaluate.
Consider expressions like \(4 \ln(x+6) - 3 \ln x\). To condense this into one term, apply the power rule initially to move numbers in front of the logs as exponents inside the logarithms. Transform \(4 \ln(x+6)\) into \(\ln((x+6)^4)\) and \(-3 \ln(x)\) into \(- \ln(x^3)\).
Next, use the quotient rule to represent subtraction between logs as division within a single log: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). So \(\ln((x+6)^4) - \ln(x^3)\) becomes \(\ln\left(\frac{(x+6)^4}{x^3}\right)\). Condensed expressions appear neater and easier to comprehend.

Applying Logarithmic Rules

In various mathematical problems, applying logarithmic rules correctly is vital for simplification and solving. When working with problems that ask for condensing or expanding logarithms, it's crucial to follow the rules methodically.
Start with identifying which property applies: Are you moving numbers as powers (power rule), adding or subtracting logs (product or quotient rules)? Apply the power rule first to set exponentials within the log, then move on to combining terms with either the product or quotient rules.
If you break a complex expression like 4 \(\ln(x+6)\) - \(3 \ln x\), first convert using the power rule to get \(\ln((x+6)^4)\) and \(- \ln(x^3)\). Then combine using the quotient rule to achieve \(\ln(\frac{(x+6)^4}{x^3})\).

• Identify logarithmic properties applicable
• Use power rule before product/quotient rules
• Simplify by reforming multiple logs into a single term

By systematically applying these rules, solving equations or transforming expressions becomes a straightforward task.