Problem 33 - College Algebra

Question

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. $$ \log _{3}(22) $$

Step-by-Step Solution

Answer

\( \log_{3}(22) \approx 2.81327 \)

Step 1: Understand the Change of Base Formula

The change of base formula allows us to calculate logarithms with any base using common logarithms (base 10) or natural logarithms (base e). The change of base formula is: \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \), where \( b \) is the base of the logarithm you want to compute, \( a \) is the argument of the logarithm, and \( c \) is the new base (usually 10 or \( e \)). In this exercise, we'll use natural logs (\( \ln \)).

Step 2: Apply the Change of Base Formula

Substitute \( a = 22 \), \( b = 3 \), and \( c = e \) into the change of base formula: \( \log_{3}(22) = \frac{\ln(22)}{\ln(3)} \). This expression now allows us to calculate the logarithm using any scientific calculator that has a \( \ln \) function.

Step 3: Calculate \( \ln(22) \)

Use a calculator to find the natural log of 22. Generally, it will look something like this: \( \ln(22) \approx 3.09104 \). Make sure that you are using the calculator correctly and rounding appropriately.

Step 4: Calculate \( \ln(3) \)

Similarly, calculate the natural log of 3: \( \ln(3) \approx 1.09861 \). Again, ensure accuracy by using your calculator properly.

Step 5: Divide to Find the Logarithm

Divide the result from Step 3 by the result from Step 4: \( \frac{3.09104}{1.09861} \approx 2.81327 \). This is the approximate value of \( \log_{3}(22) \), rounded to five decimal places.

Key Concepts

Natural Logarithms

The natural logarithm is a type of logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. Natural logarithms are represented using the shorthand \( \ln \). Their primary purpose is to simplify complex mathematical calculations related to growth models and exponential functions.

Some important properties make natural logarithms particularly useful in various fields, such as mathematics, engineering, and the natural sciences.

• The natural logarithm of 1 is 0: \( \ln(1) = 0 \)
• The natural logarithm of \( e \) is 1: \( \ln(e) = 1 \)
• The natural logarithm can transform multiplication into addition: \( \ln(ab) = \ln(a) + \ln(b) \)

Calculating the natural logarithm of a number means finding the power to which \( e \) must be raised to obtain that number. With a scientific calculator, finding \( \ln \) values becomes an easy task, and once mastered, it enhances the understanding of logarithmic relations.

Common Logarithms

Common logarithms are logarithms with base 10, commonly written as \( \log \), without specifying the base explicitly. They are often used for ease of calculation in situations involving decimal numbers, such as measurement and sound.

Their simplicity stems from using a base related to the decimal system, which aligns with our most customary numerical representations. Here are some key insights about common logarithms:

• The common logarithm of 1 is 0, because 10 to the power of 0 is 1: \( \log(1) = 0 \)
• The common logarithm of 10 is 1: \( \log(10) = 1 \)
• Similar to natural logs, you can use common logs to convert multiplication into addition: \( \log(ab) = \log(a) + \log(b) \)

Understanding common logarithms provides a strong basis for exploring more advanced logarithmic functions, where base 10 provides intuitive insights into numbers that are naturally expressed in the decimal form.

Logarithmic Functions

Logarithmic functions form a fundamental part of algebra and calculus, providing insight into behaviors of exponential growth and decay processes. A logarithmic function typically expresses the relationship \[ y = \log_b(x) \] where \( b \) is the base and \( x \) is the number for which logarithm is taken.

The base \( b \) dictates the shape of the curve that a logarithmic graph will take, as well as its practical applications. Key properties of logarithmic functions include:

• Inverse of exponential functions: Using logarithms, you can uncover the power needed to achieve a specific number using a given base.
• Logarithmic functions exhibit a unique curve that approaches the vertical axis but never touches it, known as becoming asymptotic.
• They are particularly effective for expanding the scale of numbers, making comparing large differences more manageable.

Whether in natural or common form, logarithmic functions hold various real-world applications, touching on finance, computer science, and even astronomy. Mastery of logarithmic functions enhances analytical capability across many disciplines.