Problem 68 - College Algebra

Question

In Exercises \(41-70,\) 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 without using a calculator. $$ \frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right] $$

Step-by-Step Solution

Answer

\(\ln \left[\frac{(x+6)^{\frac{5}{3}}}{x^{\frac{1}{3}} * (x-5)^{\frac{1}{6}} * (x+5)^{\frac{1}{6}}}\right]\)

Step 1: Simplify the Expression

In this step, we're simplifying the expression by applying various log properties. Starting with the original expression \(\frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right]\). First, divide each logarithmic function by \(3\): \[\frac{5}{3} \ln (x+6)-\frac{1}{3}\ln x -\frac{1}{3}\ln \left(x^{2}-25\right)\]

Step 2: Apply Logarithms Properties

Using the power rule \(\log_b(m^n) = n \log_b(m)\), the expression is further simplified to:\[\ln (x+6)^{\frac{5}{3}} - \ln x^{\frac{1}{3}} - \ln (x^{2}-25)^{\frac{1}{3}}\]Then, use the subtraction rule \(\log_b(m)-\log_b(n) = \log_b(\frac{m}{n})\) to combine these logarithms into a single term:\[\ln \left[\frac{(x+6)^{\frac{5}{3}}}{x^{\frac{1}{3}} * (x^{2}-25)^{\frac{1}{3}}}\right]\]

Step 3: Simplify the Result

Lastly, we can simplify this expression further by simplifying the argument of the logarithm inside the brackets. The result is:\[\ln \left[\frac{(x+6)^{\frac{5}{3}}}{x^{\frac{1}{3}} * (x-5)^{\frac{1}{6}} * (x+5)^{\frac{1}{6}}}\right]\]This is as far as the expression can be simplified without knowing the numerical value of \(x\).

Key Concepts

Logarithmic Expression

Understanding logarithmic expressions is essential for solving various mathematical problems. A logarithmic expression typically takes the form \( \log_b(x) \), where \(b\) is the base, and \(x\) is the argument of the logarithm—the number you are taking the log of. The expression represents the power to which the base \(b\) must be raised to obtain the number \(x\). Imagine it as asking the question, 'To what exponent must we raise \(b\) to get \(x\)?'

In the exercise, expressions like \(\frac{1}{3}[5 \ln (x+6)-\ln x-\ln (x^2-25)]\) challenge students to apply their understanding of logarithms to condense and simplify. Here, \(\ln\) represents the natural logarithm, which has the base \(e\), an irrational constant approximately equal to 2.718. Students are required to manipulate this expression using properties of logarithms, converting a complex expression into a single, more manageable logarithm.

Condensing Logarithms

The process of condensing logarithms involves combining multiple logarithmic expressions into a single logarithm. This step is quite handy when you want to simplify complex calculations or solve equations that involve logarithmic terms.

To condense logarithms, you employ several logarithm rules to combine separate logs into one. For example, you can use the product rule \(\log_b(m) + \log_b(n) = \log_b(mn)\) to combine two logs into one via multiplication. Conversely, the quotient rule \(\log_b(m) - \log_b(n) = \log_b(\frac{m}{n})\) allows you to combine logs through division.

In the given exercise, subtraction of logarithms is equivalent to the division of their arguments, effectively condensing the expression into a single logarithm. Recognizing and applying these rules correctly is crucial to mastering the art of condensing logarithms efficiently and accurately.

Logarithm Rules

There are several key logarithm rules, or properties, that make working with logarithmic expressions much simpler. These rules are instrumental in expanding or condensing logarithms and can often be used to simplify complex expressions.

A few fundamental logarithm rules include:

• The Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\), which allows you to separate a logarithm of a product into a sum of logs.
• The Quotient Rule: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), used to separate a logarithm of a quotient into a difference of logs.
• The Power Rule: \(\log_b(m^n) = n \log_b(m)\), which lets you move the exponent in the argument of the log out in front as a coefficient.

These rules are not only theoretical concepts; they are practical tools that aid in the steps of the original exercise. Through repeated utilization of these rules, you can transform the expression step by step until you achieve a simplified, condensed form—as shown in the provided solution.