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Chapter 4: Problem 44

Given that \(\log 25 \sim 1.3979, \log 250 \sim 2.3979,\) and \(\log 2500 \approx 3.3979,\) make a conjecture for an approximation of log \(25,000 .\) Then explain why this pattern continues.

Short Answer

Expert verified
The conjecture for \ \(\log 25,000 \) is approximately 4.3979. The logarithm increases by 1 when the number increases by a factor of 10.

Step by step solution

01

Observe the Pattern

Given values are \ \ \( \log 25 \sim 1.3979 \) \ \(\log 250 \sim 2.3979 \) \ \(\log 2500 \approx 3.3979 \) \ Observe the increments of 1 in the logarithm value when the number increases by a factor of 10.
02

Make a Conjecture for \log 25,000

Since \ \(\log 25 \) is approximately 1.3979 and each increment by a factor of 10 adds 1 to the logarithm, \ \(\log 25,000 \approx 4.3979 \).
03

Explain the Pattern

The logarithmic function \ \(\log_{10}(x) \) increases by 1 for each factor of 10 increase in the argument \ (x). \ For \ \(\log_{10}(25) \) to \ \log_{10}(250) \, \ the increment factor is 10, thus the logarithm increases by 1. This pattern holds as long as the base of the logarithms remains consistent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Patterns
Logarithmic patterns help us identify how the value of a logarithm changes when the argument changes. In our example, we saw that as we increased the number by a factor of 10, the logarithm increased by 1. This consistency is a key characteristic of logarithmic functions. Consider this:
  • \(\text{log}_{10}(25) \approx 1.3979\)
  • \(\text{log}_{10}(250) \approx 2.3979\)
  • \(\text{log}_{10}(2500) \approx 3.3979\)
Notice how each time the number increased by a factor of 10 (from 25 to 250 to 2500), the logarithmic value increased by exactly 1. This occurs consistently because the base of the logarithm is 10. Understanding this pattern helps us predict logarithmic values for large numbers. Predicting the logarithmic value for \(25000\) becomes easier using this observation. We see the increase follows a clear and predictable pattern.
Base 10
Logarithms can have different bases, but base 10 (\text{log}_{10}) is a common one. This is also known as the common logarithm. When we use base 10, we compare the number to multiples of 10. That is why we see predictable increments, as explained earlier. Here鈥檚 a simple breakdown:
  • \(\text{log}_{10}(10) = 1\) because 10 is 10 to the power of 1
  • \(\text{log}_{10}(100) = 2\) because 100 is 10 squared
  • \(\text{log}_{10}(1000) = 3\) because 1000 is 10 cubed
When using base 10, each multiplication by 10 increases the logarithmic value by 1. This makes calculations straightforward and consistent. In the exercise, \text{log}_{10}(25) is approximately 1.3979, and as we scaled up the number by factors of 10, the logarithm increased by 1 each time. Base 10 logarithms simplify understanding large numbers and their logarithms.
Increment by Factor of 10
Observing the provided values, we see that every time the number is increased by a factor of 10, its logarithm increases by 1. This clear and predictable pattern makes logarithmic calculations more manageable. Let鈥檚 go through a quick refresher:
  • From **25** to **250** (a factor of 10), we increase the logarithm from roughly **1.3979** to **2.3979**
  • From **250** to **2500** (again a factor of 10), the logarithm goes from **2.3979** to **3.3979**
  • So from **2500** to **25000** (another factor of 10), it鈥檚 logical to say the logarithm increases from **3.3979** to approximately **4.3979**
Each increment by a factor of 10 results in an increase of 1 in the logarithmic value. This understanding is crucial when dealing with logarithms as it helps simplify complex calculations into more predictable increments.

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Most popular questions from this chapter

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