Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch a rough graph of \(y=\ln x-\ln \left(x^{3}\right)+4 \ln \left(x^{2}\right) .\) (Hint: This will be straightforward after you have rewritten in the form \(y=K \ln x\), where \(K\) is a constant.)

Step-by-Step Solution

Answer

The function \(y = \ln x - \ln(x^{3}) + 4*\ln(x^{2})\) simplifies to \(y = 6*\ln x\). The graph of the function is an upward vertical stretch of the basic function y=ln(x) by a factor of 6.

Step 1: Simplify the Function

Use the properties of logarithms to rewrite and simplify the function. According to the properties, the difference of two logarithms is equivalent to the log of the quotient. Furthermore, the multiplication of a logarithm by a constant is equivalent to the log of the term raised to that constant. Therefore, the original function can be rewritten and simplified as follows: \(y = ln(x) - ln(x^3) + 4*ln(x^2) = ln(x) - ln(x^{3}) + ln((x^{2})^{4}) = ln(x / x^{3} * (x^{8})) = ln(x^{6})\).

Step 2: Rewrite the Function in the Form y=K*ln(x)

The function y=ln(x^6) can be further rewritten into the standard form y=K*ln(x). By using the power rule for logarithms, where the exponent a in the term ln(x^a) can be moved in front of the log, the function y=ln(x^6) can be rewritten as y=6*ln(x). Here, K equals to 6.

Step 3: Sketch the Graph

Now, sketch the graph of the function y=6*ln(x). The function y=6*ln(x) is an upward vertical stretch of the basic logarithm function y=ln(x) by a factor of 6. Start by sketching the basic function of y=ln(x), which starts at negative infinity when x approaches 0 from the right, and goes upward as x increases. Then, stretch this graph vertically by a factor of 6. It will still start at negative infinity when x approaches 0 from the right, but it will go upward more steeply as x increases.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when it comes to simplifying logarithmic expressions and handling more complex logarithmic functions. A logarithm, at its core, is the opposite of exponentiation; it asks the question: to what power must we raise a given base to obtain a certain number? For instance, if we have the logarithm \(\ln(x)\), we are using the natural logarithm, where the base is the mathematical constant \(e\).

There are several properties of logarithms that allow us to manipulate and simplify expressions:

• \textbf{Product Rule}: \(\ln(xy) = \ln(x) + \ln(y)\), which means the logarithm of a product is the sum of the logarithms.
• \textbf{Quotient Rule}: \(\ln(x/y) = \ln(x) - \ln(y)\), displaying that the logarithm of a quotient is the difference of the logarithms.
• \textbf{Power Rule}: \(\ln(x^a) = a\ln(x)\), where an exponent within a logarithm can be brought to the front as a coefficient.

These properties are not only essential for simplifying logarithmic expressions but also play a critical role in solving logarithmic equations. By mastering them, you can rewrite complex logarithmic functions into more workable forms, such as in the exercise where we combined and simplified terms to end with a constant multiplied by a single logarithm.

Power Rule for Logarithms

The power rule for logarithms is a transformation that significantly eases the process of dealing with exponential terms within logarithmic functions. The rule states that for any positive real number \(x\) and real number \(a\), \(\ln(x^a) = a\ln(x)\). This property enables us to pull exponents out in front of the log, turning a potentially complicated expression into a much simpler one.

For example, in the provided exercise, \(y = \ln(x^6)\) was transformed using the power rule to \(y = 6\ln(x)\). This not only simplifies the expression but also makes it easier to graph, as we can treat the constant factor as a stretch factor applied to the graph of \(y = \ln(x)\). This transformation is pivotal as it reduces otherwise complex logarithmic functions into simpler ones that we can easily interpret and manipulate graphically and algebraically.

Transformations of Logarithmic Functions

Graphing logarithmic functions often entails utilizing a series of transformations to the base logarithm function \(y = \ln(x)\) in order to accurately depict their behavior. Transformations include shifting, stretching, compressing, and reflecting the graph. When it comes to stretching, as we saw in the exercise, multiplying the entire function by a constant, \(K\), such as \(y = K\ln(x)\), will stretch or compress the graph vertically, depending on whether \(K\) is greater or less than 1.

In our exercise, we identified a vertical stretch by a factor of six, giving us a new function \(y = 6\ln(x)\). This means that at every point \(x\), the \(y\)-value of the base log function is multiplied by six, resulting in a steeper ascent as \(x\) increases. Conversely, if \(K\) were less than 1, the graph would be compressed and rise less steeply. Recognizing these transformations is essential when graphing logarithmic functions, as they modify the base graph's shape and position, allowing us to sketch a function accurately with relative ease.