Problem 67 - 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 \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{3}\left|2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right| $$

Step-by-Step Solution

Answer

The logarithmic expression can be condensed to \( \frac{1}{3} \ln\left( \frac{(x+5)^{2}}{x^{3}-4x} \right) \) under the condition \( \frac{(x+5)^{2}}{x^{3}-4x} > 0 \).

Step 1: Apply logarithmic properties

We begin by applying the laws of logarithms. Firstly, the subtraction of logarithms turns into a division inside the logarithm. Our expression becomes \[ \frac{1}{3} \biggl| 2\ln(x+5) - \ln\left( \frac{x}{x^{2}-4} \right) \biggr| \]. Next, the multiplication of a logarithm by a number turns into an exponent on the function inside the logarithm, resulting in \[ \frac{1}{3} \biggl| \ln((x+5)^{2}) - \ln\left( \frac{x}{x^{2}-4} \right) \biggr| \]

Step 2: Condense into a Single Logarithm

The difference of two logarithms can be written as a divide operation inside a single logarithm. This rearrangement gives us \[ \frac{1}{3} \biggl| \ln\left( \frac{(x+5)^{2}}{x(x^{2}-4)} \right) \biggr| \]

Step 3: Simplify Inside the Logarithm

We can now simplify the division inside the logarithm to reduce it further. Doing so changes the equation to \[ \frac{1}{3} \biggl| \ln\left( \frac{(x+5)^{2}}{x^{3}-4x} \right) \biggr| \]

Step 4: Apply Absolute Value

We now resolve the absolute value. As \(\ln(a)\) is defined for \( a > 0\), we can drop the absolute value and ensure that \( \frac{(x+5)^{2}}{x^{3}-4x} > 0 \) for our final answer to be valid. So, our final expression is \[ \frac{1}{3} \ln\left( \frac{(x+5)^{2}}{x^{3}-4x} \right) \]

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms, which are the inverses of exponential functions. A logarithm, typically written as \(\log_b a\) or \(\ln a\) for natural logarithms, answers the question: 'To what power must we raise the base \(b\) to obtain \(a\)?' Understanding logarithmic expressions is crucial in various fields of science and mathematics because they help us deal with exponential relationships in a more manageable form.

When working with these expressions, it is important to identify the base and the argument. For natural logarithms, the base is the mathematical constant \(e\), approximately equal to 2.71828. When you see an expression like \(\ln(x+5)\), it means the power to which \(e\) must be raised to get the value of \(x+5\). Simplifying logarithmic expressions often requires using properties of logarithms, which transform complex logs into simpler components or consolidate multiple logs into a single one.

Condensing Logarithms

Condensing logarithms means transforming a complex logarithmic expression into a single logarithm. This process is useful for solving logarithmic equations, simplifying expressions, and comparing logarithms.

To do this, we apply the properties of logarithms, such as the product rule (\(\log_b(MN) = \log_b(M) + \log_b(N)\)), the quotient rule (\(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\)), and the power rule (\(\log_b(M^p) = p\log_b(M)\)). By utilizing these rules, we can combine several logarithmic terms into one single term, making the expression easier to work with. For instance, the expression \(\ln(x+5)^2 - \ln\left(\frac{x}{x^2-4}\right)\) can be condensed into \(\ln\left(\frac{(x+5)^2}{x(x^2-4)}\right)\), transforming three terms into just one.

Laws of Logarithms

The laws of logarithms are mathematical rules that govern how to manipulate logarithmic expressions. These properties are critical when simplifying, expanding, or condensing logarithmic statements.

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\) — lets us separate the log of a product into the sum of logs.
• Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\) — lets us separate the log of a quotient into the difference of logs.
• Power Rule: \(\log_b(M^p) = p\log_b(M)\) — allows us to move the exponent in a log argument to the front as a multiplier.
• Change of Base: \(\log_b(M) = \frac{\log_k(M)}{\log_k(b)}\) — enables change between different log bases.

Employing these laws, we can transform relatively unwieldy expressions into simpler forms that we can more easily evaluate or use in further calculations.

Absolute Value in Logarithms

When incorporating absolute value with logarithmic expressions, we manage expressions that involve \(\left| \log_b(x) \right|\). Absolute value in logarithms typically indicates that we're dealing with a range of x-values that make the inside of the log positive. Since the logarithm function is only defined for positive numbers, it's critical to ensure that the argument of the log is greater than zero before removing the absolute value.

In the given solution step, the equation included the absolute value \(\frac{1}{3}\left| \ln\left( \frac{(x+5)^2}{x(x^2-4)} \right)\right|\). To drop the absolute value, we need to guarantee that the argument \(\frac{(x+5)^2}{x(x^2-4)}\) is positive. This introduces additional consideration, as it may lead to a conditional solution where we have to evaluate the range of x-values satisfy this condition. Often, this involves solving inequalities, which is a key step before the absolute value can be confidently removed from the logarithmic expression.