#properties of logarithms

[forgottenbones]

The iPhone of Slide Rules - Numberphile

[er-cryptid]

[pantheraleo04]

I just figured out a new (or rather new-to-me) property of logarithms. If you are multiplying logs together, you can swap the arguments and bases around and it will still have the same product. Here's the proof using the change of base formula:

Then, because multiplication is communicative, you can flip the order and look at this as the bases or the arguments being switched. Also if you have many logs being multiplied, you can pair up the arguments and bases in whatever way you want, which I think is pretty neat.

I wasn't sure if this still holds true once you start dealing with complex numbers because I don't have much experience in that area. But thanks to a friend giving me a crash course on complex logarithms, it appears that it does which is great.

[chemblrish]

Hi! Do you have any tips for studying chemistry? For some reason I cant seem to get all the formulas in my brain.

Hey!

My unhelpful but still favorite advice for shoving formulas into one's brain is to understand them 😅 A purely memorization-based approach is very bad for chemistry.

If the problem seems to be particularly understanding/ remembering formulas:

Ask yourself if this particular formula is just words turned into numbers and mathematical symbols. I think it may not work for everyone, but for example I found it easier to remember the literal definition of pH that is "the negative decimal logarithm of hydrogen ion concentration" rather than "pH = -log [H+]" bc otherwise I'd keep forgetting about the minus sign.

Check if you find deriving a formula from another formula easier than just memorizing it. Again, my personal example is I hate memorizing things so much I never really bothered to remember the equation that describes Ostwald's law of dilution - bc I knew I could easily, quickly, and painlessly derive it from the equilibrium constant for concentration + degree of dissociation (and I've done it so many times now it stuck in my brain anyway).

When all else fails, I turn to mnemotechnics. To this day I remember that Clapeyron's equation goes pV = nRT because many years ago someone on the internet shared a funny sentence whose words start with these 5 letters. The sillier the better.

If the issue is with chemistry in general:

Take it chapter by chapter. Chemistry, like most STEM subjects, is just blocks of knowledge upon blocks of knowledge. For example, if you want to learn electrolysis, you need to understand redox reactions first. Try to identify where the struggle begins and work from there.

Once you've picked a topic you want to work on, follow the reasoning in your textbook. If you get stuck, that might be a sign you're simply missing a piece of information from a previous chapter. If an example comes up, try to solve it along with the tips in the textbook.

If anything remains unclear, it's usually not the best idea to just leave it and move on. If the textbook becomes unhelpful, turn to the internet or maybe a friend. Otherwise, the next chapter may just turn out to be needlessly confusing.

Practice problems practice problems practice problems!! And not just the numerical ones. The theory-based ones where they ask you about reactions, orbitals, the properties of the elements etc. are important too.

Choose understanding over memorizing whenever possible.

Try to look at the big picture: the way certain concepts are intertwined, how one law may be a logical consequence of another law you learnt before, why some concepts are taught together, why you had to learn something else first to get to what you're studying now. Again, as an example, I think it's particularly fun to see towards the end of ochem, somewhere around the biomolecules: you need to integrate your knowledge of aromatic compounds, ketones and aldehydes, alcohols, carboxylic acids... Stack new information upon what you already know.

Study methods I'm a big fan of: spaced repetition, solving past papers (anything I can get my hands on tbh), flashcards for the things I absolutely have to memorize, exchanging questions and answers with a friend, watching related videos.

If by any chance you end up taking pchem, I have a post for that specifically.

I hope you can find something helpful here :) Good luck!

[joompheart]

Andrew in the Epilogue (I've got a "main story" for these guys, then a couple designs for the characters later after the final fight).

She gets just fucking mutilated in the final fight with Teddy, her entire body covered in gnarly burn and scar tissue. She takes so much damage that she is appreciably smaller when she finally heals back to functionality. The lost bone and muscle and organ tissue make her about half a head shorter and (if you can believe it) even skinnier. For anyone keeping track, she's gone from disgustingly skinny to revoltingly skinny. Her magical resilience allowed her to heal back to full functionality, but not full strength by a long shot.

More description under the cut.

Notably, the only part of her body not torn to hell is her right breast, just like Theodore. It's slightly clipped at the top but her tiny boob is otherwise untouched. The other breast was ruined along with the rest of her body.

Andrew is still strong for her size, but big chucks of flesh missing from her muscles means that she's meaningfully weaker than she was (sans magic). Her sword was also cleanly snapped by Teddy in their fight, removing the ability to modulate it's length. It's still longer than could physically fit in the sheath, but it's always that size when unsheathed. Other than that, the sword retains the rest of it's pre-snap properties, such as strength, cutting power, and ability to be magically enhanced.

Since "losing" the fight with Theodore, Andrew has stopped minmaxxing her so much and has been exploring more diverse applications for her magic abilities. This broadened focus has diminished her previous abilities, as narrow practice with magic applies mild bonuses and multipliers to those powers. This change in priority is reflected in the paler color of her tools. However, each kind of magic power has a not quite logarithmic curve for improvement/ investment. This means that to anyone but the most astute magic users, Andrew looks just as fast and deadly as she did when using her main skills. In fact, overall she's slightly more effective as she has more diverse tools to deal with any situation.

Andrew doesn't wear any sort of upper clothing at all since her armor and bodysuit were destroyed in the fight. Barring prolonged exposure to extreme circumstances like freezing weather or the vacuum of space, Andrew will never cover her body from the pubes up. Andrew now styles all of her hair into spikes, armpit and pubes included. Andrew's only sort of regular clothing are her shiny black pants, which attach to boots the same way Jotaro's do in JoJo's part 6.

"Losing" to Theodore and a harsh exchange with Jordan afterward prompted Andrew to change her demeanor and goals in life. Without anyone in the solar system who would be a meaningful enemy, much less a genuine threat, Andrew returns to her home city to rebuild in accordance with her new values.

OH YEAH EDIT::: she lost her right eye in the fight too, she’s a cyclops now.

[friends-w-ghosts]

I have an open note Precalc exam soon and finished all my notes, only to realize I have free will and am an artist and there’s nothing stopping me from drawing FFVI characters all over it.

So now I’ve got Terra reminding me what the properties of logarithms are.

[kiskisur]

Lambert W Function, W(x)

In mathematics, the Lambert W function, also called the product logarithm, is a multivalued function, namely the branches of the converse relation of the function f(w) = we^w, where w is any complex number and e^w is the exponential function.

For each integer k there is one branch, denoted by W[k](z), which is a complex-valued function of one complex argument. W[0] is known as the principal branch. These functions have the following property: if z and w are any complex numbers, then

we^w = z, which holds if and only if

w = W[k](z) for some integer k.

When dealing with real numbers only, the two branches W[0] and W[-1] suffice: for real numbers x and y the equation

ye^y = x

can be solved for y only if x ≥ −1/e.

; we get y = W[0](x) if x ≥ 0 and the two values y = W[0](x) and y = W[-1](x) if −1/e ≤ x < 0.

The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials.

For example, The maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions)

It also occurs in the solution of delay differential equations, such as y′(t) = ay(t − 1).

In biochemistry, and in particular enzyme kinetics, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.

mahal is this u..

[er-cryptid]

Properties of Common Logarithms

[noiseproblems]

Noise wall that lessen noise and look lovely

For many developers, concrete sound barriers and noise wall is a required part of their land development plan according to Ministry of Environmental Noise level guidelines.

Other than cost, one of the main challenges has been aesthetics; lessening noise to acceptable decibels has rarely taken beauty into consideration.

Thereby, beautiful subdivisions with beautiful homes and lovely industrial and commercial properties have been built, surrounded by unsightly, towering walls.

Nonetheless, new creativity in acoustic window inserts has imparted developers precisely what they have been looking for, the ability to lessen the noise pollution of owners with a wall that is also an enticing addition to the overall value of the property.

Noise reduction wall panels

Sound engineers argue that noise transfer through any void the same as water can. For noise reduction wall panels to be effective, all cracks, joints and voids should be sealed to efficiently lessen the noise transfer.

As sound levels are measured using a logarithmic scale, reduction of nine decibels is equal to eradication of almost 80% of the undesired sound.

• Reflective: Sturdy surfaces like concrete or masonry are considered to be reflective.

• Absorptive: A sound lessening noise wall with a sound muffling content and porous surface material is said to be absorptive. This implies little or zero noise is reflected back towards the source.

Double wall with air gap

Constructing a double leaf door with an air gap in between is one of the most efficient ways to soundproof a wall.

Use soundproof material that is particularly designed to block sound. In between the studs, you will want to add extra insulation. The best soundproofing insulation for insider the studs relies on different aspects, including your budget, particular requirements, and the current structure of your walls. It has better sound absorbing attributes and higher mass than regular drywall.

[thesunchronicle1]

The Impact of Quantum Computing on Modern Science and Industry

Quantum computing, a groundbreaking advancement in computational technology, has the potential to revolutionize various fields by harnessing the principles of quantum mechanics. Unlike classical computers, which use bits as the smallest unit of data, quantum computers use quantum bits, or qubits, that can represent and process information in ways that classical bits cannot. This unique capability allows quantum computers to solve complex problems exponentially faster than their classical counterparts. This article explores the impact of quantum computing on modern science and industry, highlighting its potential applications, benefits, and challenges. Joe Lemus Attleboro

Understanding Quantum Computing

Quantum computing leverages the principles of quantum mechanics, specifically superposition and entanglement, to perform computations.

Superposition: Qubits can exist in multiple states simultaneously, thanks to superposition. While a classical bit can be either 0 or 1, a qubit can be both 0 and 1 at the same time. This ability enables quantum computers to process a vast number of possibilities concurrently, significantly increasing computational power.

Entanglement: Entanglement is a phenomenon where qubits become interconnected in such a way that the state of one qubit directly influences the state of another, regardless of distance. This property allows quantum computers to perform complex calculations more efficiently by leveraging the correlation between entangled qubits.

Impact on Modern Science

Drug Discovery and Material Science: Quantum computing has the potential to revolutionize drug discovery and material science by simulating molecular and atomic interactions with unprecedented accuracy. Traditional methods of simulating these interactions are computationally intensive and often infeasible for large molecules. Quantum computers can model complex chemical reactions and predict the behavior of new drugs and materials at the quantum level, accelerating the discovery process and leading to more effective and safer drugs.

Cryptography: Quantum computing poses both challenges and opportunities for cryptography. On one hand, quantum computers can break many of the cryptographic protocols that secure current communications, such as RSA and ECC, by efficiently solving problems like factoring large integers and computing discrete logarithms. On the other hand, quantum computing also offers the potential for developing new cryptographic methods, such as quantum key distribution (QKD), which provides theoretically unbreakable encryption based on the principles of quantum mechanics.

Optimization Problems: Many scientific and engineering problems can be formulated as optimization problems, where the goal is to find the best solution from a set of possible solutions. Quantum computers are particularly well-suited for solving complex optimization problems due to their ability to explore multiple solutions simultaneously. Applications range from optimizing supply chains and traffic flow to improving financial portfolio management and energy distribution.

Climate Modeling: Quantum computing can enhance climate modeling by processing vast amounts of data and simulating complex climate systems more accurately. Improved climate models can lead to better predictions of climate change impacts, helping policymakers and scientists develop more effective strategies for mitigating and adapting to climate change.

Impact on Industry

Finance: The finance industry stands to benefit significantly from quantum computing. Quantum algorithms can optimize trading strategies, manage risk, and detect fraud more efficiently than classical algorithms. Quantum computers can also solve complex mathematical problems in financial modeling and pricing derivatives, providing more accurate and faster results.

Supply Chain and Logistics: Quantum computing can transform supply chain management and logistics by optimizing routes, inventory levels, and production schedules. Quantum algorithms can handle the complexity of supply chain networks, considering multiple variables and constraints simultaneously to find optimal solutions that minimize costs and improve efficiency.

Artificial Intelligence and Machine Learning: Quantum computing has the potential to accelerate advancements in artificial intelligence (AI) and machine learning (ML). Quantum algorithms can process large datasets and perform computations at speeds unattainable by classical computers, leading to faster training of machine learning models and improved performance in tasks such as image and speech recognition, natural language processing, and predictive analytics. Joe Lemus

Cybersecurity: As quantum computing threatens existing cryptographic systems, it also offers new solutions for cybersecurity. Quantum-resistant algorithms and quantum key distribution (QKD) provide enhanced security measures to protect sensitive information. Industries that handle critical data, such as banking, healthcare, and government, can benefit from quantum-enhanced cybersecurity to safeguard against future quantum threats.

Manufacturing: Quantum computing can optimize various aspects of manufacturing, from product design and material selection to production processes and quality control. By simulating material properties and manufacturing processes at the quantum level, companies can develop innovative products, reduce waste, and improve efficiency.

Challenges and Opportunities

Technical Challenges: Despite its promise, quantum computing faces several technical challenges that need to be addressed for widespread adoption. Qubits are highly sensitive to environmental disturbances, leading to errors and decoherence. Developing robust error correction methods and stable qubits is crucial for building practical quantum computers. Joe Lemus

Scalability: Building scalable quantum computers with a large number of qubits is a significant challenge. Current quantum computers are in the experimental stage, with limited qubits and computational capabilities. Advancements in quantum hardware and architecture are necessary to achieve scalable quantum systems that can tackle real-world problems.

Software Development: Quantum computing requires new algorithms and software tools to harness its full potential. Developing quantum algorithms that outperform classical algorithms is an ongoing research area. Additionally, programming quantum computers requires specialized knowledge, and creating user-friendly quantum software development kits (SDKs) and platforms is essential for broader adoption.

Workforce Development: The growth of quantum computing will require a skilled workforce with expertise in quantum mechanics, computer science, and related fields. Investing in education and training programs to develop quantum talent is critical for advancing quantum research and industry applications.

Ethical and Societal Implications: The advent of quantum computing raises ethical and societal questions, particularly regarding privacy and security. As quantum computers can potentially break current encryption methods, safeguarding sensitive information and developing quantum-resistant cryptography are paramount. Policymakers, technologists, and ethicists must collaborate to address these issues and ensure that quantum computing benefits society as a whole.

Quantum computing represents a paradigm shift in computational technology, with the potential to transform modern science and industry. From revolutionizing drug discovery and cryptography to optimizing supply chains and advancing artificial intelligence, the applications of quantum computing are vast and varied. However, realizing the full potential of quantum computing requires overcoming significant technical and societal challenges. By addressing these challenges and fostering collaboration among researchers, industry leaders, and policymakers, we can harness the power of quantum computing to drive innovation and create a more advanced and secure future.

[hammadchauhdary]

What to Expect from Quantum Computing in the Next Decade

Quantum computing is poised to revolutionize numerous fields, from cryptography to drug discovery, and even climate modeling. As we stand on the brink of a quantum revolution, it's essential to understand what advancements and impacts we can expect in the next decade. Here’s a detailed look into the future of quantum computing and its potential to transform various industries.

Quantum Computing: A Brief Overview

Quantum computing leverages the principles of quantum mechanics to process information in fundamentally different ways than classical computers. While classical computers use bits as the smallest unit of data, quantum computers use quantum bits or qubits. These qubits can exist in multiple states simultaneously, thanks to a phenomenon known as superposition. Additionally, qubits can be entangled, allowing them to be interconnected in ways that classical bits cannot.

Key Advancements Expected

Increased Qubit Stability and Error Correction

One of the major challenges in quantum computing today is qubit stability and error rates. Over the next decade, significant advancements are expected in quantum error correction techniques and qubit coherence times. These improvements will make quantum computers more reliable and capable of handling complex computations over longer periods.

Quantum Supremacy and Beyond

In 2019, Google claimed to have achieved quantum supremacy, where a quantum computer performed a task that would be practically impossible for a classical computer. In the coming years, we can expect more demonstrations of quantum supremacy, with quantum computers tackling increasingly complex problems that were previously unsolvable.

Scalability

Developing scalable quantum systems is crucial for their practical application. Companies like IBM, Intel, and Rigetti are working on creating quantum processors with thousands of qubits. By the end of the decade, we might see quantum computers with millions of qubits, enabling them to perform highly intricate computations.

Industries Poised for Transformation

Cryptography

Quantum computing poses a significant threat to current cryptographic techniques, such as RSA and ECC, which rely on the difficulty of factoring large numbers or solving discrete logarithms. Quantum computers could crack these codes efficiently, necessitating the development of quantum-resistant cryptography. Organizations are already working on post-quantum cryptographic algorithms to secure data against future quantum attacks.

Drug Discovery and Material Science

Quantum computers have the potential to simulate molecular and chemical reactions at an unprecedented scale, dramatically accelerating drug discovery and the development of new materials. This could lead to breakthroughs in treatments for diseases, new materials with novel properties, and more efficient industrial processes.

Artificial Intelligence and Machine Learning

Quantum computing could vastly improve machine learning algorithms by providing exponential speedups in data processing and optimization tasks. This could enhance everything from natural language processing to complex decision-making systems, making AI more powerful and efficient.

Financial Services

Quantum computing can revolutionize financial modeling, risk analysis, and portfolio optimization by processing vast amounts of data faster and more accurately than classical computers. This can lead to better investment strategies and more robust financial systems.

The Road Ahead

While the promise of quantum computing is immense, there are significant hurdles to overcome. Building practical and scalable quantum computers requires substantial advancements in hardware, software, and error correction. Moreover, developing a skilled workforce to drive these advancements is crucial.

Conclusion

The next decade will likely see quantum computing transition from a theoretical research field to a practical technology with wide-ranging applications. As we prepare for this quantum leap, staying informed about the latest developments and understanding their potential impact will be vital for businesses, governments, and individuals alike.

To delve deeper into the future of quantum computing, check out the full article here.

[qifeng618618]

Some papers coauthored with Aying Wang by Professor Dr. Feng Qi

Some papers coauthored with Aying Wang (万阿英) Aying Wan and Feng Qi, Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine, Electronic Research Archive (2024), accepted; available online at https://www.researchgate.net/publication/379643916. Feng Qi and Aying Wan, A closed-form expression of a remarkable sequence of…

[mathsminds]

Feature of Math

Mathematics

Content of Mathematics

Course Overview:

This course aims to provide a comprehensive understanding of fundamental mathematical concepts and techniques. Through a combination of theory, problem-solving exercises, and practical applications, students will develop critical thinking skills and mathematical proficiency necessary for success in higher-level mathematics and related fields.

Module 1: Number Systems

Understanding the properties of real numbers

Integers, rational numbers, irrational numbers, and their properties

Introduction to complex numbers and their operations

Exploring number patterns and sequences

Module 2: Algebraic Expressions and Equations

Simplifying algebraic expressions

Solving linear and quadratic equations

Factoring polynomials and solving polynomial equations

Graphing linear and quadratic functions

Module 3: Functions and Relations

Understanding the concept of a function

Identifying types of functions: linear, quadratic, exponential, logarithmic, etc.

Analyzing graphs of functions and their transformations

Solving systems of linear equations and inequalities

Module 4: Geometry

Exploring geometric shapes and properties

Understanding angles, lines, and polygons

Calculating area, perimeter, and volume of geometric figures

Introduction to trigonometry: sine, cosine, tangent, and their applications

Module 5: Probability and Statistics

Understanding basic concepts of probability

Calculating probabilities of events and outcomes

Introduction to descriptive statistics: mean, median, mode, and range

Analyzing data sets and making statistical inferences

Module 6: Calculus

Introduction to limits and continuity

Understanding derivatives and their applications

Calculating rates of change and optimization problems

Introduction to integrals and their applications in finding area and volume

Module 7: Discrete Mathematics

Exploring combinatorics and counting principles

Introduction to sets, relations, and functions

Understanding logic and proof technique

Exploring graph theory and its applications

Module 8: Mathematical Modeling

Understanding the process of mathematical modeling

Formulating mathematical models for real-world problems

Analyzing and interpreting mathematical models

Evaluating the effectiveness and limitations of mathematical models

Module 9: Applications of Mathematics

Exploring interdisciplinary applications of mathematics in science, engineering, finance, and other fields

Case studies and real-world examples demonstrating the relevance of mathematical concepts

Ethical considerations and implications of mathematical applications

Module 10: Review and Final Assessment

Reviewing key concepts and techniques covered in the course

Solving comprehensive problem sets and practice exam

Final assessment covering all topics and skills learned throughout the course.

[math-journal]

Algebra 1 — here — Next

Intermediate Algebra

Cheat Sheet

Rational Expression Properties, Exponential Properties, Radical and Fractional Properties, Complex Numbers, Quadratic Equations, Function Operations, Exponential and Logarithmic Functions, Parental Functions

UNIT 1

Functions

Introduction

Vertical Line Test

Translating Function aka shifting

Domain / Range

Interval Notation

Radicals

Basic

Simplify / Multiply

Addition/Subtraction

Foil/Factor Solving

F*ck off

Imaginary

Imaginary/Complex Numbers

Solving with Imaginary Numbers

Problem #1

Problem #2

Note: sometimes I write side-to-side rather than up-and-down while solving certain questions for ease. By math conventions, this is frowned upon. Stick to vertical orientation

Unit 1 REVIEW

.

Unit 2

Factoring Trinomial (R)

Sum of Square - Difference of Square (R)

2nd Degree Trinomial

3rd Degree Trinomial

Completing the Square

Quadratic Formula

Quadratic Function (parabola)

Flipping shortcut

Graphing Quadratic Functions

Standard to Vertex Form

Overview

Unit 2 Review

.

Unit 3

Decay and Growth Functions

Intro to Log + Properties

Common & Natural Log

Application Problems

More examples

3 Ways Solve Logs

Say Shit Outloud

Complicated Log / In

Log Property Cheat Sheet

Log Application Prob

Log Tricks

Domain/Range Fraction F(x)

Restrictions

Fractions + Restrictions

More Application

Canceling Review

Unit 3 Review

.

Unit 4

Review

Function Operations

.