Week 5 Discussion Question: How Many Integers From 1 Through

Week 5 Discussion Questionfor How Many Integers From 1 Through 10 Can

For how many integers from 1 through 10 can the natural logarithms be approximated given that ln2 ≈ 0.6931, ln3 ≈ 1.0986, and ln5 ≈ 1.6094? Use the laws and rules of logarithms to approximate these logarithms (DO NOT USE A CALCULATOR). Explain your answers.

Paper For Above instruction

The natural logarithm function, denoted as ln(x), is a fundamental concept in mathematics, particularly in calculus, logarithmic, and exponential functions. Understanding how to approximate natural logarithms using the laws and properties of logarithms is essential, especially when calculators are unavailable or when developing an intuitive grasp of the behavior of logarithmic functions. In this discussion, we analyze which integers from 1 through 10 can have their natural logarithms approximated using the given values and the laws of logarithms.

Understanding Logarithm Laws and Given Data

The problem provides specific values for ln(2), ln(3), and ln(5), which serve as reference points for approximations of other logarithms. The laws of logarithms—product, quotient, and power rules—are instrumental in approximating the natural logs of other numbers within the specified range. These laws are as follows:

• Product Rule: ln(ab) = ln(a) + ln(b)
• Quotient Rule: ln(a/b) = ln(a) - ln(b)
• Power Rule: ln(a^k) = k * ln(a)

Using these, we can attempt to approximate ln(1) through ln(10).

Approximations of Logarithms for Numbers 1–10

Let's start with the known values and derive approximations for the remaining integers.

ln(1):

Since ln(1) = 0, this is exact. It serves as a baseline for the other calculations.

ln(2):

Given directly as 0.6931.

ln(3):

Given directly as 1.0986.

ln(4):

Recognizing that 4 = 2^2, apply the power rule: ln(4) = 2 ln(2) ≈ 2 0.6931 = 1.3862.

ln(5):

Given directly as 1.6094.

ln(6):

Using the product rule: 6 = 2 * 3, so ln(6) ≈ ln(2) + ln(3) ≈ 0.6931 + 1.0986 = 1.7917.

ln(7):

Since 7 is not directly related to the known values, we can approximate using known values or interpolate. Recognizing that 7 is between 6 and 8, we consider 8 = 2^3, so ln(8) = 3 ln(2) ≈ 3 0.6931 = 2.0794. Since 7

ln(8):

As previously calculated, ln(8) = 3 * ln(2) ≈ 2.0794.

ln(9):

Since 9 = 3^2, apply the power rule: ln(9) = 2 ln(3) ≈ 2 1.0986 = 2.1972.

ln(10):

10 = 2 * 5, so ln(10) = ln(2) + ln(5) ≈ 0.6931 + 1.6094 = 2.3025.

Summary of Approximations

Conclusion

By leveraging the laws of logarithms, we successfully approximated the natural logarithms of integers 1 through 10 without a calculator, relying instead on known reference values and fundamental properties. Most of these approximations are consistent with the actual values, illustrating how mathematical laws can facilitate estimations and deepen understanding of logarithmic behavior. These approximations are valuable in contexts where precise calculations are infeasible and demonstrate the power of logarithmic rules in practical and theoretical applications.

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