Problem 19 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{x}{4} $$

Step-by-Step Solution

Answer

The expanded logarithmic expression is \( \ln(x) - 1.38629436112\).

Step 1: Understanding the Initial Logarithmic Expression

Firstly, identify the form of the given expression. The given logarithmic expression is \( \ln \frac{x}{4} \), which is the natural logarithm (ln) of a division or quotient. It expresses that the base e, raised to the power of this logarithm, equals the given argument.

Step 2: Applying the Quotient Rule

Next, we can use the Quotient Rule of logarithms to simplify. The Quotient Rule states that the logarithm of a quotient equals the difference of the logarithms. It can be written as \(\ln(a/b) = \ln(a) - \ln(b)\). When we apply the Quotient Rule to our expression, we get \( \ln(x/4) = \ln(x) - \ln(4) \).

Step 3: Simplify the Expression

Lastly, the \(\ln(4)\) in the expression is a constant, as \(\ln(4) \approx 1.38629436112\). So, we have the expanded logarithmic expression as \( \ln(x) - 1.38629436112\).

Key Concepts

Natural Logarithm

The natural logarithm, known commonly as "ln," is a special type of logarithm. It uses the constant \(e\) as its base, where \(e\approx 2.71828\). This constant is an irrational number, much like \(\pi\), and is very important in mathematics, especially in calculus and exponential growth problems.
When you see the notation \(\ln(x)\), it is asking "to what power must \(e\) be raised to yield \(x\)?" Using natural logarithms makes it easier to work on growth processes or decay that continuously occur over time.

• \(\ln(1) = 0\): Because \(e^0 = 1\).
• \(\ln(e) = 1\): Because \(e^1 = e\).
• \(\ln(e^x) = x\): This captures the essence of what a logarithm does by reversing the exponential function.

Whenever you see the natural logarithm in equations or word problems, remember its base is always \(e\), helping transition between different levels of scale easily.

Quotient Rule

The Quotient Rule for logarithms is a useful property when dealing with divisions within a logarithmic expression. It states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.
This rule is applied as follows:

• When you have an expression \( \ln\left(\frac{a}{b}\right) \), it can be decomposed into \( \ln(a) - \ln(b) \).
• This is particularly useful for breaking down complex expressions into simpler components.

For example, in the exercise \( \ln\left(\frac{x}{4}\right) \), you apply the quotient rule to expand this into \( \ln(x) - \ln(4) \). This helps in simplifying or solving logarithmic equations and understanding their underlying behavior.
Remember that the quotient rule doesn't change the equality between expressions; it only rephrases them to allow for easier computation and further manipulation.

Logarithm Properties

Understanding the properties of logarithms is like having a toolbox for simplifying and solving logarithmic problems. These properties make it possible to rewrite complex logarithms in a more manageable form.
Here are some of the key properties:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n\log_b(m) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \(c\) is a new base of your choice.

In the original exercise, the Quotient Rule was heavily utilized to expand \( \ln\left(\frac{x}{4}\right) \) into \( \ln(x) - \ln(4) \). By knowing these properties, you can tackle various logarithm problems with confidence and flexibility, converting them into forms that are easier to compute or interpret conceptually.