Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Approximate \(\log _{3} 16\) (with error less than \(0.005\) ) using your calculator. (b) Rewrite \(\log _{3} 16\) in terms of \(\log\) base 10 . (c) Rewrite \(\log _{3} 16\) in terms of log base \(e\). (d) Rewrite \(\log _{3} 16\) in terms of log base 7 .

Step-by-Step Solution

Answer

(a) Approximately 2.632 (to three decimal places). \n(b) \(\log \_3 16 = \log \_10 16 / \log \_10 3\). \n(c) \(\log \_3 16 = ln 16 / ln 3\). \n(d) \(\log \_3 16 = \log \_7 16 / \log \_7 3\).

Step 1: Approximation

For part (a), use the calculator to approximate \(\log _{3} 16\). Make sure to set the error margin to less than 0.005.

Step 2: Change of Base

For part (b), use the change of base formula to rewrite \(\log _{3} 16\) in terms of log base 10. The formula is \(\log \_b a = \log \_c a / \log \_c b\). So, here \(\log \_3 16 = \log \_10 16 / \log \_10 3\).

Step 3: Using Natural Logarithm

For part (c), use the same change of base formula to rewrite \(\log \_3 16\) in terms of log base e (natural logarithm). So the equivalent will be \(\log \_3 16 = ln 16 / ln 3\).

Step 4: Logarithm Base Change

For part (d), rewrite \(\log \_3 16\) in terms of log base 7. This can be achieved by using the change of base formula \(\log \_b a = \log \_c a / \log \_c b\). Hence, \(\log \_3 16 = \log \_7 16 / \log \_7 3\).

Key Concepts

Change of Base Formula

The change of base formula is a handy tool when dealing with different logarithm bases. It allows you to convert a logarithm from one base to another, which is especially useful if your calculator only computes logs in certain bases (like base 10 or the natural logarithm base, e). The general formula is given by:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

This means you can calculate \( \log_b a \) by dividing \( \log_c a \) by \( \log_c b \), regardless of the base \( c \) you choose. This approach simplifies the process of calculating logarithms by using a consistent method and a set of known values.
For example, to find \( \log_3 16 \) in terms of base 10, we use:

• \( \log_3 16 = \frac{\log_{10} 16}{\log_{10} 3} \)

Similarly, you can rewrite \( \log_3 16 \) using the natural log base e (ln):

• \( \log_3 16 = \frac{\ln 16}{\ln 3} \)

And for any other base, like base 7, the process remains the same.

Approximation

Approximation is a technique used to find a value that is close to the true answer, often used when an exact number isn't necessary or isn't available. In the context of logarithms, calculators can give us an approximate result that is usually adequate for most practical purposes.
The problem at hand requires finding \( \log_3 16 \) to an error less than 0.005. Using a calculator, you can compute this by applying the change of base formula and rounding off the result as needed. Remember, approximation does not mean inaccuracy, but a controlled method of providing a number close to the actual value while considering a small margin of error. This way, results are both manageable and sufficiently precise for real-world applications.

Logarithm Bases

Logarithm bases define the number to which you are raising your power to achieve a given number in a logarithmic equation. Logarithms can be used in any positive base, and different bases can be more or less suited to particular applications.
The most common ones are:

• Base 10 (common logarithms) noted as \( \log_{10} \), often used in scientific calculations and engineering.
• Base e (natural logarithms, noted as \( \ln \)), vital in calculus and many areas of higher mathematics, as well as natural phenomena.
• Any other base you might find useful for specific purposes, like base 2 in computer science.

Understanding various bases helps in flexible problem-solving and utilizing the right tools for the job at hand. This flexibility is where the change of base formula shines, allowing seamless conversion between these varying bases.

Calculus Concepts

In calculus, logarithms play a crucial role, especially when dealing with exponential growth and decay, as well as integrals and derivatives. The natural logarithm base, e, is particularly special in calculus due to its mathematical properties.
For example, the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). This simple form makes \( \ln \) a strong candidate for ease of computation in calculus equations. Furthermore, logarithmic differentiation is a method used to simplify the differentiation of products and quotients of functions.
Understanding how to manipulate logarithms, using tools like the change of base formula, significantly benefits calculus students. It not only helps in solving algebraic equations but also paves the way to tackle more complex calculus problems efficiently. As you get comfortable with these concepts, tackling related calculus problems becomes less daunting.