Discovering Logarithms: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Introduction

From my earliest memories, I have been surrounded by numbers and patterns. The world presented itself as a canvas of quantities—counting toys, measuring time, and observing growth. As I interacted with my environment, I began to notice relationships between numbers that went beyond simple counting. Through curiosity and logical reasoning, I embarked on a journey to understand these relationships more deeply.

In this narrative, I will detail how, starting from basic experiences as an infant, I independently observed, experimented, and logically deduced the concept of logarithms. This journey illustrates how complex mathematical ideas can emerge from simple observations and reasoning, without relying on subjective definitions or prior formal education.

Chapter 1: Exploring Numbers and Multiplication1.1 Early Encounters with Counting and AdditionCounting Objects

• Activity: Counting my toys—1, 2, 3, ..., up to 10 or more.

• Observation: Each additional toy increases the total count by one.

Understanding Addition

• Example: Combining 2 apples and 3 apples to get 5 apples.

• Semantics: Addition as the process of combining quantities.

1.2 Discovering MultiplicationRepeated Addition

• Observation: Adding the same number multiple times yields a larger total.

• Example: $2+2+2=6$

• Reflection: There is a more efficient way to represent repeated addition.

Defining Multiplication

• Concept: Multiplication as repeated addition.

• Example: $2\times 3=6$ represents adding 2 three times.

• Semantics: Multiplication simplifies the expression of repeated addition.

Chapter 2: Understanding Powers and Exponents2.1 Observing Growth PatternsDoubling Quantities

• Experiment: Folding a piece of paper in half repeatedly.

• Observation: Each fold doubles the number of layers.

• Sequence: 1 layer, 2 layers, 4 layers, 8 layers, etc.

Exponential Growth

• Observation: Doubling leads to rapid increases in quantity.

• Reflection: Recognizing a pattern where quantities grow by a constant factor.

2.2 Introducing ExponentsExpressing Repeated Multiplication

• Concept: Exponents represent repeated multiplication of the same number.

• Example: 23=2×2×2=82^3 = 2 \times 2 \times 2 = 823=2×2×2=8

• Semantics: The exponent indicates how many times the base is multiplied by itself.

Exploring Exponential Sequences

• Sequence of Powers of 2:

• 20=12^0 = 120=1

• 21=22^1 = 221=2

• 22=42^2 = 422=4

• 23=82^3 = 823=8

• 24=162^4 = 1624=16

• Observation: Each term is obtained by multiplying the previous term by 2.

Chapter 3: Encountering Inverse Problems3.1 Asking Reverse QuestionsFrom Multiplication to Division

• Question: If $2\times 3=6$, what number multiplied by 2 gives 6?

• Answer: 3, because $6÷2=3$.

• Semantics: Division is the inverse operation of multiplication.

From Powers to Roots

• Question: If 23=82^3 = 823=8, what is the number that when raised to the power of 3 gives 8?

• Answer: 2, because 83=2\sqrt[3]{8} = 238​=2.

• Semantics: Roots are the inverse operation of exponentiation.

3.2 Seeking Inverse Functions for ExponentsIdentifying the Need for a New Concept

• Observation: I can easily compute 23=82^3 = 823=8, but how do I find the exponent if I know the result?

• Question: What exponent $x$ satisfies 2x=82^x = 82x=8?

• Answer: $x=3$, since 23=82^3 = 823=8.

• Challenge: For non-integer exponents, such as solving 2x=102^x = 102x=10.

Chapter 4: Introducing Logarithms4.1 Defining LogarithmsConceptualizing the Logarithm

• Definition: The logarithm is the inverse operation to exponentiation.

• Notation: log⁡ba=x\log_b a = xlogb​a=x means bx=ab^x = abx=a.

• Semantics: The logarithm answers the question, "To what exponent must the base be raised to produce a given number?"

Understanding Through Examples

• Example 1: log⁡28=3\log_2 8 = 3log2​8=3, because 23=82^3 = 823=8.

• Example 2: log⁡10100=2\log_10 100 = 2log1​0100=2, because 102=10010^2 = 100102=100.

4.2 Calculating LogarithmsUsing Known Powers

• Method: Use known exponents to find logarithms.

• Example: To find log⁡216\log_2 16log2​16:

• Since 24=162^4 = 1624=16, log⁡216=4\log_2 16 = 4log2​16=4.

Estimating Non-Integer Logarithms

• Challenge: Finding log⁡210\log_2 10log2​10.

• Approach:

• Recognize that 23=82^3 = 823=8 and 24=162^4 = 1624=16.

• Since $10$ is between $8$ and $16$, log⁡210\log_2 10log2​10 is between $3$ and $4$.

Chapter 5: Exploring Logarithmic Properties5.1 Logarithm LawsProduct Rule

• Statement: log⁡b(mn)=log⁡bm+log⁡bn\log_b (mn) = \log_b m + \log_b nlogb​(mn)=logb​m+logb​n.

• Example: log⁡2(8×4)=log⁡28+log⁡24\log_2 (8 \times 4) = \log_2 8 + \log_2 4log2​(8×4)=log2​8+log2​4.

• log⁡232=5\log_2 32 = 5log2​32=5, since 25=322^5 = 3225=32.

• log⁡28=3\log_2 8 = 3log2​8=3, log⁡24=2\log_2 4 = 2log2​4=2.

• Sum: $3+2=5$, confirming the rule.

Quotient Rule

• Statement: log⁡b(mn)=log⁡bm−log⁡bn\log_b \left( \frac{m}{n} \right) = \log_b m - \log_b nlogb​(nm​)=logb​m−logb​n.

• Example: log⁡2(84)=log⁡28−log⁡24\log_2 \left( \frac{8}{4} \right) = \log_2 8 - \log_2 4log2​(48​)=log2​8−log2​4.

• log⁡22=1\log_2 2 = 1log2​2=1, since 21=22^1 = 221=2.

• Difference: $3-2=1$, confirming the rule.

Power Rule

• Statement: log⁡b(mk)=k⋅log⁡bm\log_b (m^k) = k \cdot \log_b mlogb​(mk)=k⋅logb​m.

• Example: log⁡2(23)=3⋅log⁡22\log_2 (2^3) = 3 \cdot \log_2 2log2​(23)=3⋅log2​2.

• log⁡28=3\log_2 8 = 3log2​8=3, log⁡22=1\log_2 2 = 1log2​2=1.

• Product: $3\times 1=3$, confirming the rule.

5.2 Understanding Logarithmic ScalesApplications in Measurement

• Example: The Richter scale for earthquakes is logarithmic.

• Semantics: Each whole number increase represents a tenfold increase in amplitude.

Chapter 6: Applying Logarithms to Real-World Problems6.1 Doubling Time and Half-LifeExponential Growth and Decay

• Observation: Certain quantities grow or decay exponentially over time.

• Example: Population growth, radioactive decay.

Calculating Time Using Logarithms

• Problem: How long does it take for a population to double if it grows at a constant rate?

• Solution: Use logarithms to solve 2=ert2 = e^{rt}2=ert, where $r$ is the growth rate.

6.2 Solving for Exponents in EquationsExample Problem

• Equation: 5x=1255^x = 1255x=125.

• Solution:

• Recognize that 125=53125 = 5^3125=53.

• Therefore, $x=3$, since 5x=535^x = 5^35x=53.

Using Logarithms for Non-Integer Solutions

• Equation: 3x=203^x = 203x=20.

• Solution:

• Take logarithms on both sides: log⁡3x=log⁡20\log 3^x = \log 20log3x=log20.

• Apply power rule: $x\cdot \log 3=\log 20$.

• Solve for $x$: x=log⁡20log⁡3x = \frac{\log 20}{\log 3}x=log3log20​.

Chapter 7: Deepening Understanding of Logarithms7.1 Change of Base FormulaDeriving the Formula

• Statement: log⁡ba=log⁡kalog⁡kb\log_b a = \frac{\log_k a}{\log_k b}logb​a=logk​blogk​a​, where $k$ is any positive value.

• Understanding:

• Allows computation of logarithms with any base using a known logarithm function.

Example:

• Calculate: log⁡210\log_2 10log2​10 using common logarithms (log⁡10\log_{10}log10​).

• Solution:

• log⁡210=log⁡1010log⁡102=1log⁡102\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{\log_{10} 2}log2​10=log10​2log10​10​=log10​21​.

7.2 Natural LogarithmsUnderstanding $e$ and $\ln$

• Number $e$: An irrational constant approximately equal to 2.71828.

• Natural Logarithm: ln⁡x=log⁡ex\ln x = \log_e xlnx=loge​x.

• Semantics: Natural logarithms arise naturally in continuous growth processes.

Applications:

• Calculus: Differentiation and integration involving exponential and logarithmic functions.

• Growth Models: Modeling populations, interest rates, and other natural phenomena.

Chapter 8: Reflecting on the Discovery8.1 The Power of Inverse Operations

• Insight: Understanding inverses is key to solving equations involving exponentials.

• Semantics: Logarithms provide a means to "undo" exponentiation.

8.2 Recognizing the Ubiquity of Logarithms

• Applications:

• Science: pH in chemistry, decibels in acoustics.

• Technology: Data compression, algorithms in computer science.

• Reflection: Logarithms are a fundamental tool across various fields.

Conclusion

Through observation, experimentation, and logical reasoning, I was able to discover and understand the concept of logarithms. Starting from basic counting and multiplication, I explored exponents and recognized the need for an inverse operation to solve exponential equations. By evolving the semantics of each concept explicitly and grounding them in reality, I grasped logarithms as the inverse of exponentiation.

This journey highlights how complex mathematical ideas can emerge naturally from simple experiences. It demonstrates that with curiosity and logical thinking, foundational knowledge about mathematics can be accessed and understood without relying on subjective definitions or advanced prior knowledge.

Logarithms opened up a new realm of problem-solving and understanding of the natural world, providing powerful tools to deal with exponential relationships and scales. This exploration deepened my appreciation for the interconnectedness of mathematical concepts and their practical applications in everyday life.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational mathematical principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to recognize patterns, formulate hypotheses, and test them against observations, we can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of mathematical relationships.

Note: This detailed narrative presents the conceptualization of logarithms as if I, an infant, independently observed and reasoned them out. Each concept is derived from basic experiences, emphasizing the natural progression from counting and multiplication to the understanding of logarithms. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about mathematics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".