Problem 5 - Thomas Calculus

Question

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } \log _{3} x} & {\text { b. } \ln 2 x} \\\ {\text { c. } \ln \sqrt{x}} & {\text { d. } \sqrt{x}} \\ {\text { e. } x} & {\text { f. } 5 \ln x} \\ {\text { g. } 1 / x} & {\text { h. } e^{x}}\end{array} $$

Step-by-Step Solution

Answer

\( \log_3 x, \ln 2x, \text{ and } 5 \ln x \) grow at the same rate as \( \ln x \). \( \ln \sqrt{x} \text{ and } 1/x \) grow slower. \( \sqrt{x}, x, \text{ and } e^x \) grow faster.

Step 1: Analyze Similar Growth Functions

Consider the function \( \log_{3} x \). Since \( \log_{3} x = \frac{\ln x}{\ln 3}\), it grows at the same rate as \( \ln x \) because the difference is just a constant factor. \( \ln 2x = \ln x + \ln 2\) also has the same growth rate as \( \ln x \) for large \( x \). Finally, observe \( 5 \ln x \), which grows at the same rate as \( \ln x \) because it is a constant multiple, affecting only the coefficient but not the growth rate.

Step 2: Identify Slower Growing Functions

Analyze \( \ln \sqrt{x} \). This can be rewritten as \( \ln x^{1/2} = \frac{1}{2} \ln x \), indicating it grows slower than \( \ln x \) since it is half the original growth rate. Similarly, \( \frac{1}{x} \) approaches zero as \( x o \infty \), so it grows slower (actually, it decreases, not grows).

Step 3: Evaluate Faster Growing Functions

For \( \sqrt{x} \), consider that it grows faster than \( \ln x \) because eventually \( \sqrt{x} > \ln x \) for very large \( x \) due to the power law growing faster than logarithmic growth. The function \( x \) grows even faster because a linear function will always surpass a logarithmic function for large \( x \). Finally, \( e^x \) grows faster than any polynomial or logarithmic function, including \( \ln x \), due its exponential growth rate.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fascinating part of mathematics. These functions help us understand the growth patterns and rate of various types of phenomena.
- At its core, a logarithmic function like \ \( \ln x \ \) is the inverse of an exponential function, making it especially useful in problems where we need to comprehend very rapid growth or decay. A key property is that its growth becomes slower as \( x \) increases, resembling a curve that gently approaches infinity.
- For example, if we consider \( \log_{3} x \), through the relationship \( \log_{3} x = \frac{\ln x}{\ln 3} \), we see that its rate of growth is aligned with \( \ln x \), albeit each point is scaled down by the constant factor \( \ln 3 \). This means that both functions will have the same behavior in terms of growth rate.
- Logarithmic functions are substantially slower compared to many other functions such as polynomial or exponential functions. This property of logarithmic functions makes them particularly useful in fields like computer science where efficiency and resource management are crucial.

Exponential Functions

Exponential functions exhibit a type of growth that is incredibly rapid and expansive. Unlike logarithmic functions which grow slowly, exponential functions can quickly move towards infinity.
- The function \( e^x \) is a classic example of an exponential function. It grows much faster than a logarithmic function like \( \ln x \). In fact, exponential functions will outgrow any polynomial function for sufficiently large values of \( x \).
- The reason for this fast growth lies in their structure. While logarithmic functions multiply the input by a constant factor, exponential functions raise the number to the power of the input, causing the output to increase dramatically.
- This characteristic of exponential functions makes them critical in modeling processes that have rapid change, such as radioactive decay, population growth, or compound interest scenarios.

Comparing Function Growth Rates

Understanding how different functions grow relative to each other helps in selecting the right models for different scenarios.
- Let's compare already discussed logarithmic functions and exponential functions with other common functions, such as polynomials or square roots. The function \( \sqrt{x} \) grows faster than \( \ln x \) because even though both start very gradually, \( \sqrt{x} \) increasingly overtakes \( \ln x \) due to its square root nature.
- A linear function like \( x \) will inevitably surpass \( \ln x \) since it increases indefinitely with a constant slope, far exceeding the diminishing returns of the logarithm.
- Meanwhile, \( e^x \) doesn't just surpass \( \ln x \); it leaves it far behind due to its rapid exponential growth.
- On the slower side, functions like \( \ln \sqrt{x} \) or \( 1/x \) fail to keep up with the pace of \( \ln x \). The former even mirrors half the growth rate of \( \ln x \), while \( 1/x \) doesn't grow at all—it actually tends towards zero as \( x \rightarrow \infty \).
In summary, by comparing growth rates, we see a clear hierarchy: \( 1/x < \ln \sqrt{x} < \ln x \equiv \log_3 x < \sqrt{x} < x < e^x \). This ranking aids in effectively choosing the function when dealing with real-life data.