Problem 33 - College Algebra with Corequisite Support

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

\(\log_3(22) \approx 2.81359\).

Step 1: Understanding the Change-of-Base Formula

The change-of-base formula is used to evaluate logarithms with any base. It states that \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \) for any positive base \( c \). We often use a natural log (\( \ln \)) or common log (\( \log \)) for base \( c \).

Step 2: Apply the Change-of-Base Formula

Using the change-of-base formula, replace \( b \) with 3 and let \( a \) be 22. We choose the natural log as the common base, so the expression is transformed to:\[\log_3(22) = \frac{\ln(22)}{\ln(3)}\]

Step 3: Calculate Natural Logs

Using a calculator, find the natural log of 22 and the natural log of 3. Ensure your calculator is in the correct mode for natural logarithms.- \( \ln(22) \approx 3.09104 \)- \( \ln(3) \approx 1.09861 \)

Step 4: Evaluate the Expression

Substitute the calculated values from Step 3 into the expression derived from the change-of-base formula:\[\log_3(22) = \frac{3.09104}{1.09861} \approx 2.81359\]Use a calculator to perform this division and obtain the result.

Key Concepts

change of base formula

The change of base formula is a very helpful tool in mathematics, enabling us to evaluate logarithms that might otherwise seem complex. It allows any logarithm to be expressed with a base of our choice. Imagine you're dealing with a logarithm like \( \log_b(a) \) and you find it tricky because the base \( b \) doesn't easily align with your calculator's common functions, which usually handle base 10 or the natural base, \( e \) (Euler's number). With the change of base formula, you can rewrite your logarithm in a more manageable form. The formula is:

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

This simply means, you can take the log of \( a \) and \( b \) using a new base \( c \) of your choice, commonly using base 10 or, as often chosen, the natural base \( e \). The result is the same as your original logarithm. Remember, the use of \( c \) is optional, but using familiar bases like 10 or \( e \) makes calculations more convenient, especially when using a calculator.

natural logarithms

Natural logarithms are particularly interesting because they use Euler's number \( e \), which is roughly 2.71828, as a base. You will see natural logs denoted as \( \ln(x) \) instead of \( \log_e(x) \). They are extensively used in various fields such as mathematics, physics, and engineering due to their unique properties and simplicity. The use of the natural logarithm simplifies many equations because \( e \) is the base of the natural exponential function. This creates a kind of symmetry between logarithmic and exponential operations.

• Natural logarithms express growth processes very conveniently, from population growth to radioactive decay.
• They're key to solving equations involving exponential growth by transforming multiplicative processes into additive ones.

When evaluating expressions with natural logarithms, especially with the help of a calculator, make sure to use the \( \ln \) button to directly compute logs with base \( e \). This makes them particularly straightforward for calculations involving the change of base formula, as seen in \( \log_3(22) = \frac{\ln(22)}{\ln(3)} \).

calculator usage

Using a calculator to evaluate logarithms is a straightforward process if you follow some simple guidelines. Most scientific calculators have dedicated buttons for both common logarithms (\( \log \))—which uses base 10—and natural logarithms (\( \ln \))—which use base \( e \). Here’s how to effectively use them:

• Ensure your calculator is in the correct mode if it requires switching between modes; usually, this isn't necessary for logs but check any specific instructions.
• For natural logarithms, simply locate and press the \( \ln \) button, enter your number, and press Equal or Enter to get your result.

In our example of \( \log_3(22) \), you would: - First, press \( \ln \), type 22, and jot down or remember the result, approximately 3.09104. - Then, repeat the process for 3, getting approximately 1.09861. - Finally, input the division \( \frac{3.09104}{1.09861} \) into your calculator to find the approximate value, resulting in 2.81359. Using a calculator is all about making sure you're comfortable with accessing these functions and following through with the operations step-by-step to ensure precision.