Problem 34 - Algebra for College Students

Question

Approximate each logarithm to three decimal places. $$ \log _{2} 93 $$

Step-by-Step Solution

Answer

\(\log_2 93 \approx 6.538\).

Step 1: Understand Logarithm Basics

Recall the definition of logarithms: \(\log_b a = c\) means that \(b^c = a\). In this problem, we want to find \(x\) such that \(2^x = 93\).

Step 2: Apply Change of Base Formula

To approximate \(\log_2 93\), use the change of base formula: \(\log_2 93 = \frac{\log_{10} 93}{\log_{10} 2}\). This allows us to use a calculator which typically handles base 10 logarithms.

Step 3: Calculate Individual Logarithms

Use a calculator to find \(\log_{10} 93\) and \(\log_{10} 2\). These are approximately 1.968 and 0.301 respectively.

Step 4: Divide the Logarithms

Divide the results from Step 3: \(\frac{1.968}{0.301} \approx 6.538\).

Step 5: Round the Result

Round this division result to three decimal places. The approximate value of \(\log_2 93\) is 6.538.

Key Concepts

Logarithm Basics

Logarithms can be initially confusing, but they're quite simple once you understand the core idea. Essentially, a logarithm answers the question: "To what power must a certain base be raised to yield a certain number?" For example, if you have an expression like \( \log_b a = c \), it means that when the base \( b \) is raised to the power \( c \), it equals \( a \).
In the exercise provided, you deal with \( \log_2 93 \). This specifically asks: "What power must 2 be raised to so the result is 93?" Understanding this question provides the foundation for further manipulation and approximation when working with logarithms.

Change of Base Formula

The change of base formula is incredibly useful when you need to find logarithms for bases not commonly computed. Basic scientific calculators usually handle only base 10 or base \( e \) logarithms. This formula allows flexibility.

• The formula is \( \log_b a = \frac{\log_c a}{\log_c b} \).
• Using this to find \( \log_2 93 \), it becomes \( \frac{\log_{10} 93}{\log_{10} 2} \).

By converting the logarithm to a base like 10, you leverage your calculator’s functionality to compute logs not directly available. It serves as a bridge to accessing any base logarithm through familiar bases.

Calculator Use in Logarithms

Calculators make logarithmic calculations more accessible, especially when applying concepts like the change of base formula. To approximate \( \log_2 93 \), follow these steps:

• First, find \( \log_{10} 93 \) by entering 93 into your calculator and using the \( \log \) function. This gives approximately 1.968.
• Next, find \( \log_{10} 2 \). Input 2 and press the \( \log \) function to get approximately 0.301.
• Finally, divide these two results: \( \frac{1.968}{0.301} \) results in an approximate value of 6.538.

Rounding this division to three decimal places, as specified in problems, helps ensure precision. Calculators, hence, streamline the process, turning what could be complex manual calculations into straightforward steps.