Problem 34 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln y-\ln 2+\ln \sqrt{x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\ln(y\sqrt x/2)\)

Step 1: Apply logarithmic properties

First, apply the properties of logarithms, which say that the difference of two logs is equal to the log of a quotient and the sum of two logs is the log of a product. This gives a new equation: \(\ln(y/2)+\ln(\sqrt x)\)

Step 2: Simplify further

Combine the two terms by converting the sum of logs into a product: \(\ln((y/2)\sqrt x)\).

Step 3: Fix the expression inside the log to the simplest form

Simplify the expression \((y/2)\sqrt x = y\sqrt x/2\). So, the final expression is \(ln(y\sqrt x/2)\)

Key Concepts

Logarithm Simplification

Simplifying logarithmic expressions can often make complex problems easier to handle. In essence, the process involves using rules of logarithms to consolidate multiple log terms into a single expression. Consider a typical exercise like \(\ln y - \ln 2 + \ln \sqrt{x}\). The key here is to recognize that logarithmic operations have inverse processes similar to multiplication and division.

First, by subtracting logarithms \(\ln y - \ln 2\), we utilize the property \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\). This tells us that the difference of two logs with the same base is the log of the quotient of their arguments. Similarly, when adding logarithms such as \(\ln y + \ln \sqrt{x}\), we use the property \(\log_b(m) + \log_b(n) = \log_b(mn)\), meaning the sum of two logs is the log of their product.

Thus, the given expression simplifies to a single logarithm as \(\ln\left(\frac{y\sqrt{x}}{2}\right)\). Here, we have applied these properties to transform the separate terms into one more manageable logarithmic expression.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with the base \(e\), where \(e\) is Euler's number, approximately equal to 2.71828. This constant arises naturally in mathematics and is the base rate of growth shared by all continually growing processes. Natural logarithms are widely used in many fields, including science, engineering, and economics, because they simplify complex calculations involving exponential growth and decay.

The rules for simplifying expressions with natural logarithms are the same as those for logarithms with any other base. In the context of our previous exercise, you typically encounter natural logarithms when dealing with exponential functions that have a base of \(e\). The natural logarithm functions as a tool to 'undo' exponential growth, which can be highly beneficial when solving equations or simplifying expressions in calculus.

Properties of Logarithms

Understanding the properties of logarithms is crucial for simplification and problem-solving. There are several key properties that are commonly used:

• Product Rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\), which allows you to split a log of a product into a sum of logs.
• Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\), this lets you transform a log of a quotient into a difference of logs.
• Power Rule: \(\log_b(m^n) = n\log_b(m)\), which is helpful in taking out the exponent from inside the log and making it a coefficient.
• Change of Base Rule: \(\log_b(m) = \frac{\log_k(m)}{\log_k(b)}\), which is used if you need to convert from one base to another.

Each of these properties helps with transforming and combining logarithmic expressions so they can be managed more easily. For instance, simplifying the expression in our exercise involved the use of the product and quotient rules to combine three logarithmic terms into one single log expression, which not only looks cleaner but is also often closer to a form that is necessary for further calculations or applications.