Problem 55 - Algebra A Combined Function

Question

Approximate each logarithm to four decimal places. $$ \log _{4} 9 $$

Step-by-Step Solution

Answer

\( \log_4 9 \approx 1.5849 \).

Step 1: Understanding the Logarithm

The problem is asking us to approximate the logarithm of 9 with base 4 to four decimal places. Specifically, we need to find \( x \) such that \( 4^x = 9 \).

Step 2: Using the Change of Base Formula

To find \( \log_4 9 \), use the change of base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \). Common bases \( c \) typically used are 10 or \( e \), so we'll use base 10 here. The expression becomes \( \log_4 9 = \frac{\log_{10} 9}{\log_{10} 4} \).

Step 3: Calculate Individual Logarithms

Use a calculator to find \( \log_{10} 9 \) and \( \log_{10} 4 \). This gives us approximately \( \log_{10} 9 \approx 0.9542 \) and \( \log_{10} 4 \approx 0.6021 \).

Step 4: Perform the Division

Now, divide the two results: \( \frac{0.9542}{0.6021} \approx 1.5849 \).

Step 5: Round the Answer

The division result gives us \( \log_4 9 \approx 1.5849 \). Since the problem asks for four decimal places, make sure the final answer is rounded as shown.

Key Concepts

Change of Base Formula

Understanding logarithms is much easier with the change of base formula. This powerful tool lets us express logarithms with any base using common logarithms or natural logarithms, which calculators handle easily. Suppose you have a logarithm with base \( b \) and you want to convert it to a more familiar base, such as 10 (common logarithm) or \( e \) (natural logarithm). You can simply use: \[ \log_b a = \frac{\log_c a}{\log_c b} \] Here, \( c \) is your new base, typically 10 or \( e \). This formula is especially useful because:

• Most calculators readily compute \( \log_{10} \) and \( \ln \) (base \( e \)).
• It simplifies complex problems.
• It makes approximations straightforward.

So, when you're faced with hard-to-handle bases, this formula is your go-to method. Just plug in your values, and watch the magic happen.

Approximation of Logarithms

Approximating logarithms means finding value estimates instead of exact calculations. This process is often needed because calculators or even manual calculations can't always provide precise values for every base. With the change of base formula, we can easily switch to a base that calculators understand. For example, when calculating \( \log_4 9 \), we used the change of base to convert it into a form the calculator can approximate: \[ \log_4 9 = \frac{\log_{10} 9}{\log_{10} 4} \] By calculating the common logarithms of 9 and 4 separately, we got:

• \( \log_{10} 9 \approx 0.9542 \)
• \( \log_{10} 4 \approx 0.6021 \)

Then, dividing these, we can approximate: \[ \frac{0.9542}{0.6021} \approx 1.5849 \] Using this approach not only helps avoid manual errors but also provides reliable estimates quickly. It's a skill useful for mathematics students and practitioners.

Mathematical Calculations

Mastering basic mathematical calculations using logarithms can enhance one's arithmetic prowess. Steps taken in computing \( \log_4 9 \) demonstrate key arithmetic processes. First, let's cover individual logarithms. Calculators provide logarithmic values swiftly, so entering \( \log_{10} 9 \) yields roughly 0.9542, and for \( \log_{10} 4 \), approximately 0.6021. Next is division, a crucial mathematical operation. Performing \( \frac{0.9542}{0.6021} \) yields approximately 1.5849. This division step translates the logarithmic operation into a tangible number that's simple to comprehend. Let's not forget precision. Logarithms are often needed to several decimal places for accuracy. Throughout these steps:

• Ensure correct entry in calculators.
• Double-check division steps.
• Take care to round to the needed precision.

These components are vital pieces of the larger puzzle that is mathematical calculations, underscoring the importance of attention to detail and clarity when working through logarithms.