Problem 44 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 2.75$$

Step-by-Step Solution

Answer

To evaluate \( \log _{3} 2.75 \) using the change-of-base formula and a calculator, you should firstly apply the change-of-base formula, yielding \( \frac{\log_{10}(2.75)}{\log_{10}(3)} \). Then, calculate this expression using a calculator to get the final result and make sure to round to 4 decimal places.

Step 1: Understand the change-of-base formula

The formula to changes the base of a logarithm is \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). This essentially means that we can change the base of the logarithm to any base \(c\) we prefer. For this case, since we are using a calculator, we will choose either base 10 or base \(e\), as those are common bases calculators can handle. In this case, we'll use base 10.

Step 2: Apply the change-of-base formula to \(\log _{3} 2.75\)

We can substitute \(b = 3\) and \(a = 2.75\) into our formula to get \(\log _{3}(2.75) = \frac{\log_{10}(2.75)}{\log_{10}(3)}\)

Step 3: Calculate the new values

Now that we have our new formula, we can use a calculator to compute : \(\frac{\log_{10}(2.75)}{\log_{10}(3)}\). Remember to round your answer to four decimal places.

Key Concepts

Understanding Logarithms

Logarithms are a mathematical way of representing the power to which a number, called the base, must be raised to produce a given number. In simpler terms, when we say \( \log_b(a) = x \), we mean that the base \( b \) needs to be raised to the power \( x \) to get \( a \). For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).
Logarithms are helpful because they allow us to download large multiplication and power operations into simpler additions and multiplications.

• The base of a logarithm determines the scales on which the logarithm operates.
• Common bases include 10, known as the common logarithm, and \( e \) (approximately 2.718), which is known as the natural logarithm.

Understanding the concept of logarithms is crucial for solving many problems in mathematics and in applications involving exponential relationships.

Base Conversion in Logarithms

Base conversion using the change-of-base formula allows us to evaluate logarithms with different bases using more familiar bases that calculators can compute. This is crucial because most calculators only have functions for base 10 (common logarithm) and base \( e \) (natural logarithm).
The formula for base conversion is:

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

This means the logarithm \( \log_b(a) \) can be converted into any base \( c \), commonly base 10 or base \( e \). By using this formula, evaluating complex bases becomes straightforward:

• 1. Substitute the values of \( a \) and \( b \) into the formula.
• 2. Choose an appropriate base \( c \), typically 10 or \( e \), for computation.

This versatility is why base conversion is useful, particularly when inputting into a calculator.

Calculator Usage for Logarithms

Using a calculator to evaluate logarithms involves a few straightforward steps, especially when dealing with non-standard bases. Modern calculators have dedicated logarithm buttons which simplify this process:

• Most scientific calculators have \( \log \) and \( \ln \) buttons for calculating common logarithms (base 10) and natural logarithms (base \( e \)).

To compute a logarithm like \( \log_3(2.75) \), first apply the change-of-base formula to write it in terms of base 10:
\( \log_3(2.75) = \frac{\log_{10}(2.75)}{\log_{10}(3)} \)
Now, you can perform the following steps:

• Input \( \log(2.75) \) into your calculator and note the result.
• Input \( \log(3) \), also recording this output.
• Divide the first result by the second.
• Make sure to round the final answer to four decimal places for precision.

These steps ensure an accurate conversion and evaluation using the tool capabilities of a calculator.