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Question Use the change-of-base formula to approximate \( \log _{3} 24 \). (Round to 3 decimal places.)

Ask by Blake Byrd. in the United States
Dec 06,2024

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The approximate value of \( \log_{3} 24 \) is 2.895.

Solution

To approximate \( \log_{3} 24 \) using the change-of-base formula, we can use the following formula: \[ \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \] where \( k \) can be any positive number (commonly 10 or \( e \)). Here, we will use base 10 for convenience. Applying the change-of-base formula, we have: \[ \log_{3} 24 = \frac{\log_{10} 24}{\log_{10} 3} \] Now, we need to calculate \( \log_{10} 24 \) and \( \log_{10} 3 \). Using a calculator: \[ \log_{10} 24 \approx 1.3802 \] \[ \log_{10} 3 \approx 0.4771 \] Now, substituting these values into the formula: \[ \log_{3} 24 \approx \frac{1.3802}{0.4771} \approx 2.895 \] Rounding to three decimal places, we find: \[ \log_{3} 24 \approx 2.895 \] Thus, the approximate value of \( \log_{3} 24 \) is \( \boxed{2.895} \).

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To approximate \( \log _{3} 24 \), you can use the change-of-base formula, which states that \[ \log_b a = \frac{\log_k a}{\log_k b} \] for any base \( k \). Here, you can choose \( k \) as 10 or \( e \). Using base 10, we have: \[ \log_{3} 24 = \frac{\log_{10} 24}{\log_{10} 3} \] Calculating this using a calculator, you find \( \log_{10} 24 \approx 1.380 \) and \( \log_{10} 3 \approx 0.477 \). Thus, \[ \log_{3} 24 \approx \frac{1.380}{0.477} \approx 2.895. \] So, rounding to three decimal places, \( \log_{3} 24 \approx 2.895 \). Want to dive a bit deeper? Logarithms are not just mathematical abstractions; they have a rich history! The concept of logarithms was developed in the early 17th century by mathematicians such as John Napier, who wanted to simplify complex calculations. Logarithms turned multiplication into addition, making computations much easier before the age of calculators! As for practical applications, logarithms are widely used in various fields such as sound intensity (decibels), the pH scale in chemistry, and measuring earthquake magnitudes (Richter scale). So, the next time you hear about the “log scale,” you’ll know it’s not just a math thing—it’s got real-world implications too!

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