Logarithms, a key concept tested on the SAT, play a crucial role in understanding exponential growth and decay, and solving logarithmic expressions. They help evaluate logarithmic functions and simplify expressions using logarithmic laws. Logarithms also have practical applications in fields like acoustics (decibel scale), chemistry (pH scale), astronomy (star brightness), and modeling real-world phenomena where exponential relationships exist.
Mastering Logarithms: Unlocking the Secrets of Growth, Decay, and Beyond
Hey there, math enthusiasts! Welcome to our logarithmic adventure where we’ll dive into the fascinating world of functions that make exponential growth and decay look like a piece of cake. Let’s start with the basics:
1. The Definition and Key Properties of Logarithmic Functions
Logarithmic functions are the inverse of exponential functions, which means they undo what exponential functions do. They’re like the mathematical undo button!
- Definition: Logarithmic functions have the form f(x) = log(base a)x, where a is a positive number not equal to 1.
- Key Properties:
- Inverse Relationship: If y = a^x, then x = log(base a)y.
- Identity: log(base a)a = 1 for any positive number a.
- Product Rule: log(base a)xy = log(base a)x + log(base a)y
- Quotient Rule: log(base a)(x/y) = log(base a)x – log(base a)y
- Power Rule: log(base a)x^n = n log(base a)x
Now that we’ve got the basics down, let’s venture into the exciting world of exponential growth and decay!
Explain the inverse relationship between exponential and logarithmic functions.
Logarithmic Functions: The Flip Side of Exponential Growth and Decay
If exponential functions are like the rocket-fueled engines of growth and decay, then logarithmic functions are their trusty sidekicks, keeping a watchful eye on their escalating or diminishing trajectories. Think of it like this: when you’re counting your riches or watching a radioactive element disintegrate, it’s like you’re riding the rollercoaster of exponential functions. But if you jump off that rollercoaster to measure how far you’ve traveled, that’s where logarithms come in, like a trusty compass guiding you back to the origin.
The logarithmic function is like a mirror image of its exponential counterpart. For every exponential function that “blasts off” with a certain base, its logarithmic counterpart “comes back down” with the exact same base. It’s like the inverse relationship between height and distance. If an exponential function starts from a point and shoots upwards, its logarithmic buddy starts from the same point and plummets downwards.
This inverse relationship is one of the keystones of understanding logarithmic functions. It’s like a two-way street where exponential growth and decay meet logarithmic evaluation. By understanding this connection, you can seamlessly switch between these two mathematical worlds, unlocking a powerful tool for modeling real-world phenomena. So, the next time you encounter a logarithmic function, remember its exponential alter ego lurking just around the corner, ready to provide the inverse perspective.
Logarithmic Functions: Unlocking the Secrets of Growth and Decay
Imagine a world where everything grows or decays exponentially. A plant suddenly sprouts into a skyscraper, while a flashlight’s battery drains in a flash! That’s where logarithmic functions come in, like superhero mathematicians that tame these wild growth patterns.
How Logarithms Control Growth and Decay
Logarithmic functions are the cool kids that love to hang out with exponential functions. They’re like the inverse of each other, meaning if an exponential function makes something grow like a beanstalk, a logarithmic function will bring it back down to Earth.
For example, if an algorithm doubles the number of likes on your social media post every hour, that’s exponential growth. But to find out how long it took to reach a certain number of likes, you need to use a logarithmic function. It’ll tell you how many hours it took for the algorithm to work its magic.
Real-World Examples of Growth and Decay
Logarithmic functions aren’t just some math trick. They’re actually super useful in the real world. For instance:
- Epidemics: When a virus spreads, the number of infected people can grow exponentially. Logarithmic functions help scientists model this growth and predict how long it will take to contain the outbreak.
- Radioactive Decay: The amount of radioactive material decreases exponentially over time. Log functions let scientists calculate how long it will take for a sample to become safe.
- Music Fading: Turn up the volume on your headphones and the sound intensity will increase exponentially. Logarithmic functions help engineers design speakers that produce sound that decays gradually.
Logarithmic functions are the power tools of mathematics, giving us the ability to understand and control exponential growth and decay. They’re like the traffic cops of the math world, keeping everything from social media likes to viruses in check. So, next time you see a trendline that looks like a rocket or a melting ice cube, remember that logarithmic functions have got you covered!
Understanding Logarithms: Demystifying the Inverse of Exponents
In the world of mathematics, there’s a hidden gem called logarithmic functions. These functions are like the secret code that unlocks the mysteries of exponential growth and decay.
Think of it this way: exponential functions show how things grow or decay rapidly, like a viral video or a radioactive substance. Logarithmic functions, on the other hand, are like the inverse detectives that tell us how much time it took for something to grow or decay to a certain level.
Real-World Tales of Exponential Growth and Decay
Imagine you have a little ball that grows twice its size every minute. That’s exponential growth! To figure out how long it took to grow to a certain size, we need to call in the logarithmic function. It’s like a time-traveler that takes us back to the moment when the ball was just a tiny speck.
Another real-world example is the decay of radioactive substances. Uranium, for instance, loses half of its radioactivity every 4.5 billion years. So, if we want to know how old a fossilized dinosaur bone is, we can measure its remaining radioactivity and use a logarithmic function to calculate its age. How cool is that?
Demonstrate techniques for evaluating logarithmic functions and solving logarithmic equations.
Unveiling the Magic of Logarithms: A Crash Course
Hey there, math enthusiasts! Let’s dive into the fascinating world of logarithms and discover their astonishing powers.
Understanding Logarithms: The Gateway to Exponential Relationships
Logarithms are like secret codes that unlock the secrets of exponential functions. Remember those sneaky equations with bases raised to unknown exponents? Well, logarithms are their superpower, allowing us to find those elusive exponents with ease.
Exploring Exponential Growth and Decay: From Bacteria to Radioactive Elements
Logarithmic functions are like detectives solving the mystery of exponential growth and decay. They can tell us how fast bacteria multiply or how quickly radioactive elements disintegrate. From population booms to the decay of fossils, logarithms reveal the hidden patterns behind these fascinating phenomena.
Evaluating Logarithmic Expressions: Cracking the Code
Now, let’s get down to brass tacks! Evaluating logarithmic functions is akin to breaking a puzzle. We use tricks like rewriting expressions in exponential form and applying logarithmic laws to simplify those pesky equations. It’s like being a math ninja, decoding secrets with every move.
Solving Logarithmic Equations: The Quest for the Missing Exponent
What’s even cooler than evaluating logarithms? Solving logarithmic equations! It’s like a hidden treasure hunt, where we’re searching for the missing exponent. By isolating the logarithmic term and applying our knowledge of logarithms, we can unveil the mystery and find that elusive number.
Real-World Applications: Where Logarithms Shine
Logarithms aren’t just confined to textbooks; they’re like secret agents operating in the real world. From measuring the intensity of sound to determining the acidity of liquids and even calculating the brightness of stars, logarithms are everywhere, making our lives easier and more interesting.
Graphing Logarithmic Functions: Visualizing the Magic
Graphs of logarithmic functions are like artistic masterpieces that reveal the secrets of their mathematical equations. They have unique shapes and characteristics that tell us important information about the behavior of the function. From logarithmic growth to decay curves, these graphs paint a vivid picture of the underlying mathematical relationships.
Discuss the logarithmic laws and their applications in simplifying expressions.
Logarithms: The Inverse of Exponents That Unleash the Power of Simplification
When it comes to mathematics, fear not, for logarithms are our allies, not our enemies. They’re the superheroes of simplification, the secret weapon to conquer complex expressions. Let’s break down these logarithmic laws and show you how they’re your ticket to mathematical mastery.
Product Rule:
Imagine you have a product of two numbers, let’s say 3 and 4. When you take the logarithm of this product, it’s like taking a shortcut. Instead of logging both numbers and multiplying the results, you can simply add their logs. In other words, log(3 x 4) = log(3) + log(4).
Quotient Rule:
Now, let’s say you’re dealing with a fraction, like 12 divided by 3. The logarithmic quotient rule lets you subtract the log of the denominator from the log of the numerator. So, log(12/3) = log(12) - log(3). It’s like finding the difference between their logarithmic values.
Power Rule:
But wait, there’s more! If you have an expression like 5 to the power of 3, the logarithmic power rule comes into play. It allows you to bring the exponent down as a coefficient in front of the log. In this case, log(5^3) = 3 * log(5). It’s like multiplying the log of the base by the exponent.
These logarithmic laws are like the secret code to expressing complex numbers in a simpler, more manageable way. They help us conquer algebraic challenges, solve equations like a mathematical ninja, and unravel the mysteries of logarithmic graphs. So, next time you’re faced with a logarithmic expression, don’t panic. Embrace the power of these laws and simplify your way to mathematical glory!
The Decibel Scale: Measuring Sound Intensity with Logarithms
Have you ever wondered how we measure the loudness of sounds, like the deafening roar of a jet engine or the gentle whisper of a breeze? Enter the decibel scale, a logarithmic wonder that lets us quantify sound intensity with ease.
Imagine a loud concert that gives you a ringing in your ears. The sound waves hitting your eardrums are so strong that they can physically damage them. Now, think about a quiet forest stroll. The sound waves there are barely even noticeable.
The decibel scale helps us compare these vastly different sound levels by converting their intensity into a logarithmic value. Logarithms are a mathematical tool that allows us to compress a wide range of numbers into a smaller, more manageable scale.
In the decibel scale, the reference point is the threshold of human hearing, which is set at 0 decibels (dB). As sound intensity increases, the decibel value also increases, but not linearly. Instead, it follows a logarithmic pattern.
For example, a sound that is 10 times more intense than the threshold of hearing is not 10 dB, but 20 dB. And a sound that is 100 times more intense is not 100 dB, but 40 dB. This logarithmic compression means that even tiny changes in sound intensity can result in significant decibel differences.
The decibel scale is incredibly useful in various fields, from audio engineering to environmental noise monitoring. Engineers use it to design sound systems and measure noise levels in factories. Environmentalists use it to assess noise pollution and protect wildlife.
So, next time you’re at a concert or enjoying the peace of nature, remember the decibel scale. It’s the logarithmic tool that helps us measure and understand the myriad sounds that fill our world.
Unlocking the Secrets of Logarithms: From Decibels to Starry Nights
If you’ve ever wondered how scientists measure the deafening roar of a rock concert or the faint shimmer of distant stars, the answer lies in a magical mathematical tool called logarithms. Prepare to dive into the fascinating world of these wonder-working functions that help us make sense of the vast and often mind-boggling scale of our universe.
** decibels: When Sound Gets Seriously Loud**
Imagine a rock concert that’s so loud, it makes your ears ring for days. How do scientists measure this deafening experience? Enter decibels (dB), a logarithmic scale that quantifies sound intensity. Here’s where logarithms come into play: they compress the incredibly wide range of sound intensities into a manageable scale, which makes it easier for us to understand and compare different sounds.
For instance, a whisper has a decibel level of around 30 dB, while a jet engine’s thunderous roar can reach a staggering 140 dB. By using logarithms, we can express this immense difference in sound intensity in a way that makes sense to our mortal ears.
pH Scale: Measuring the Acidity or Alkalinity of a Solution
Now, let’s venture into the realm of chemistry. The pH scale is a logarithmic measure that tells us how acidic or alkaline a solution is. It ranges from 0 to 14, with 7 being neutral. Acids have a pH below 7, while bases have a pH above 7.
Logarithms step in again to help us understand this scale. By compressing the vast range of possible hydrogen ion concentrations (which determine acidity or alkalinity), logarithms make it easier for us to grasp and compare different pH values. For example, a solution with a pH of 3 is ten times more acidic than a solution with a pH of 4. How’s that for a logarithmic superpower?
Star Brightness: From Twinkling Lights to Blazing Supernovae
Last but not least, let’s journey among the stars. Astronomers use a logarithmic scale to measure the brightness of stars. This scale, known as the apparent magnitude scale (don’t let the fancy name scare you), helps us comprehend the vast differences in stellar luminosity.
The brighter the star, the lower its apparent magnitude. For example, the faintest stars visible to the naked eye have an apparent magnitude of around 6, while the brightest stars can have apparent magnitudes as low as -1. A star with an apparent magnitude of -1 is about 2.5 times brighter than a star with an apparent magnitude of 0.
And there you have it, my fellow logarithmic explorers! From decibels to pH scales and star brightness, logarithms are the unsung heroes, helping us make sense of the incredible diversity and grandeur of our world. So, next time you’re rocking out to your favorite band or testing the pH of your aquarium water, remember the magic of logarithms, the mathematical tool that turns the complex into the comprehensible.
Logarithms: Unlocking the Secrets of Growth, Decay, and the World Around Us
Understanding Logarithmic Functions
Logarithms are like secret codes that can unlock the mysteries of the universe. They’re tightly intertwined with exponential functions, which describe all sorts of growth and decay patterns we see in everyday life.
Exploring Exponential Growth and Decay
Growth and decay are like two sides of the same coin. Bacteria multiplying, radioactive elements decaying—these are all examples of exponential patterns that can be modeled with logarithmic functions.
Evaluating and Solving Logarithmic Expressions
Just like detectives solve mysteries, we can solve logarithmic expressions using special tricks and rules. These “logarithmic laws” let us simplify complex expressions into simpler ones.
Real-World Applications of Logarithms
Logarithms aren’t just for math geeks. They’re hiding in plain sight all around us:
4.2. The pH Scale: A Measure of “Acidy-ness”
In chemistry, the pH scale is like a magic potion that tells us how acidic or alkaline a solution is. Logarithms play a crucial role in measuring pH, helping us determine whether a substance is bleach-like or as sweet as honey.
Logarithms: The Secret Tool for Measuring Acidity and Alkalinity
Hey there, math enthusiasts! Today, we’re diving into the intriguing world of logarithms and exploring their sneaky but oh-so-useful role in chemistry. You know that pH scale thing you’ve heard of? The one that tells you how acidic or alkaline a solution is? Well, logarithms are the secret sauce behind it all.
Imagine this: you’ve got a lemon sitting on your kitchen counter, and you’re curious about how acidic its juice is. You grab your trusty pH meter, dip it in, and… ta-da! You get a number between 0 and 14. But what does that number really mean?
Enter logarithms. They’re like the magic wands of math that can transform numbers into something much more manageable. In the case of pH, logarithms help us understand the concentration of hydrogen ions, which determine how acidic or alkaline a solution is.
The pH scale is logarithmic, meaning it’s based on the logarithm of the hydrogen ion concentration. The lower the pH number, the higher the hydrogen ion concentration, and the more acidic the solution. The higher the pH number, the lower the hydrogen ion concentration, and the more alkaline the solution.
Logarithms help us see the big picture of hydrogen ion concentration. For example, a pH of 3 means that the hydrogen ion concentration is 10 times higher than a pH of 4. A pH of 1 means that the hydrogen ion concentration is 100 times higher than a pH of 3. It’s a ** logarithmic scale**, so each number represents a tenfold difference in concentration.
So, there you have it! Logarithms are the unsung heroes of chemistry, helping us unravel the mysteries of acidity and alkalinity. The next time you’re wondering why the lemon juice is making your tea taste so sour, just remember: it’s all thanks to the power of logarithms!
Logarithmic Scales: The Key to Unlocking Star Brightness
Hey there, number lovers! Let’s venture into the fascinating world of logarithms and their role in understanding the brightness of stars. It’s like unlocking a secret code that reveals the secrets of celestial bodies!
When astronomers gaze at the night sky, they don’t just admire the beauty of the stars; they also measure their brightness. But how do you compare the brightness of two stars that might be light-years apart? Enter the logarithmic scale, a tool that helps us make sense of the vast range of star brightnesses.
Imagine a scale where each step represents a tenfold increase in brightness. So, a star that’s 10 times brighter than another will be one step higher on the scale. And, guess what? This logarithmic scale is used to measure star brightness!
This clever scale allows astronomers to compare the brightness of stars within the same magnitude range. Magnitude refers to how bright a star appears from Earth. The lower the magnitude, the brighter the star. So, a star with a magnitude of 1 is brighter than a star with a magnitude of 5.
Fun Fact: The brightest star in the night sky, Sirius, has a magnitude of -1.46. That’s like saying it’s nearly 100 times brighter than the faintest star we can see with our naked eyes!
Understanding the logarithmic scale is crucial for astronomers because it allows them to measure and compare the brightness of stars across vast distances, helping us comprehend the incredible diversity of our cosmic neighborhood.
Logarithms: The Stellar Comparison Tool
4.3. Star Brightness
Stars twinkle in the night sky, but how do we compare their brilliance? Enter logarithms, the astronomer’s secret weapon.
Just like us, stars come in all shapes and sizes. Some shine so brightly that they outshine their celestial neighbors, while others are so dim that they’re barely visible to the naked eye. To measure this brightness, astronomers use a logarithmic scale called the apparent magnitude.
Think of the apparent magnitude as a cosmic dimmer switch. The brighter the star, the lower its apparent magnitude. So, a star with an apparent magnitude of 0 is incredibly bright, while a star with an apparent magnitude of 10 is relatively faint.
Now, here’s the logarithmic twist: a difference of 5 in apparent magnitude corresponds to a factor of 100 in brightness! This means that a star with an apparent magnitude of 5 is 100 times brighter than a star with an apparent magnitude of 10.
So, when astronomers want to compare the brightness of two stars, they simply subtract their apparent magnitudes. The absolute value of the result tells them how many times brighter or fainter one star is than the other. How cool is that?
Explain the shape and key features of graphs of logarithmic functions.
Logarithmic Functions: An Adventure into the Magical World of Graphs
Think of a logarithmic function as a sneaky secret agent, hiding in plain sight. It’s like a mirror image of its exponential counterpart, revealing truths about growth and decay. And just like a secret agent, logarithmic graphs have their own unique quirks and characteristics that can help us unravel puzzles in the world around us.
Imagine a logarithmic graph as a mysterious portal leading to a hidden realm. The x-axis is like the entrance, where values of the independent variable slyly peek through. The y-axis, on the other hand, is a magical ladder, guiding us up and down the graph as the logarithm transforms numbers into (whisper it) exponential superpowers.
Key Features of Logarithmic Graphs:
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Asymmetrical Shape: Logarithmic graphs are like lopsided dancers, with a graceful curve on one side and a straight line on the other.
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Vertical Asymptote: These graphs have a special vertical line that they always avoid, no matter how close they come. This line is called a vertical asymptote and it marks the forbidden zone where the logarithm becomes undefined.
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Intercept: Logarithmic graphs make a grand entrance at the y-axis, proudly displaying their intercept. This intercept tells us the starting point of the curve or line.
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Reflection: Logarithmic graphs are like mirror images of exponential graphs, reflecting each other’s shapes and characteristics.
Unveiling the Secrets of Logarithmic Graphs:
Logarithmic graphs unlock the hidden mysteries of exponential growth and decay. They help us understand how things increase or decrease over time, whether it’s the spread of a contagious disease or the cooling of a hot object. By studying the shape and features of logarithmic graphs, we can decipher patterns and make predictions about the future, like master detectives solving a complex crime.
Logarithmic Graphs: Unveiling the Secrets of Real-World Patterns
Logarithmic functions aren’t just mathematical marvels—they’re also the key to unlocking hidden patterns in the real world. Think of them as detective glasses that help us see exponential growth, sound intensity, and even star brightness in a whole new light.
When we graph logarithmic functions, we dive into a world where exponential patterns take on a new form. The graphs have a signature “S” shape, reflecting the rapid increase or decrease seen in exponential growth and decay.
For instance, let’s say we’re tracking the population of bunnies in a field. Their numbers might double every month, showing an exponential increase. If we plot this growth on a logarithmic graph, we’ll see a straight line, with the slope indicating the rate of growth. Cool, right?
But wait, there’s more! Logarithmic graphs can also help us determine the rate of decay. Imagine a radioactive element that loses half its mass every hour. Its decay pattern will show up as a straight line with a negative slope on a logarithmic graph.
So, the next time you encounter a logarithmic graph, don’t be intimidated. Put on your detective glasses and let the graph guide you through the hidden patterns in the world around you.
Logarithms: Beyond the Math Jargon
What Are Logarithms, Anyway?
Imagine you have a superpowered calculator that can undo exponents. That’s what logarithms are! They’re like the “anti-exponents” that can tell you what number was raised to a certain power.
Exponential Growth and Decay: The Ups and Downs of Life
Logarithms are the key to understanding how things grow and decay exponentially. Think about how a virus spreads or how money doubles when you invest it. These patterns are like roller coasters, with ups and downs that can be described by logarithmic functions.
Solving Logarithms: A Puzzle with a Secret Key
Mastering logarithms is like becoming a codebreaker. We have special logarithmic laws that are like rules for solving these tricky puzzles. With these laws, you can simplify expressions and even find solutions to logarithmic equations.
Logarithms in the Real World: Where They Shine
Logarithms aren’t just math nerd fodder; they’re super useful in the real world. Let’s dive into some epic examples:
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Decibels: Measuring Sound Intensity – Ever wondered how loud your music is? The decibel scale uses logarithms to measure sound intensity. It’s like a sound-measuring superpower!
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pH Scale: Acids and Bases – Chemists use logarithms to create the pH scale, which tells us how acidic or alkaline a solution is. It’s the key to understanding everything from your skincare routine to brewing the perfect cup of coffee.
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Star Brightness: A Cosmic Scale – Astronomers use logarithms to measure the brightness of stars. It helps them understand the vastness of our universe and compare the brilliance of celestial bodies.
Graphing Logarithms: Mapping the Ups and Downs
Logarithmic functions have unique graphs that look like a mirror image of exponential functions. They have a vertical asymptote and a horizontal intercept, which are like landmarks that help us interpret their behavior.
Modeling with Logarithms: Math in the Real World
Logarithmic functions aren’t just abstract concepts; they’re powerful tools for modeling real-world phenomena. They can describe everything from population growth to radioactive decay. By understanding logarithms, we can better understand the world around us.
Unlocking the Power of Logarithmic Functions: A Mathematical Tale
Get ready for a mathematical adventure as we delve into the fascinating world of logarithmic functions! These functions are like secret codes that can translate exponential growth and decay into patterns we can understand.
Exposing the Secrets of Logarithmic Functions
Logarithmic functions are the inverses of exponential functions, making them a duo that dances hand-in-hand. They possess some nifty properties:
- Any number to the power of 0 equals 1.
- The logarithm of 1 to any base equals 0.
- The logarithm of a product equals the sum of the logarithms of the individual numbers.
- The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.
Unveiling Exponential Growth and Decay
Logarithmic functions are like detectives that can uncover patterns in exponential growth and decay. Exponential growth happens when something increases by the same percentage over time, like a snowball rolling down a hill. Exponential decay, on the other hand, is when something decreases by the same percentage over time, like a ball bouncing up and down.
Taming Logarithmic Expressions
Evaluating and solving logarithmic expressions is like solving a puzzle. Techniques like changing bases or using logarithmic laws are your tools to crack the code. These laws include:
log(a^b) = b * log(a)log(a * b) = log(a) + log(b)log(a/b) = log(a) - log(b)
Logarithms in the Real World
Logarithms are like secret agents, working behind the scenes in a variety of applications:
Decibel Scale: They measure sound intensity, helping us understand how loud things are.
pH Scale: They determine the acidity or alkalinity of a solution, crucial in chemistry and biology.
Star Brightness: They help astronomers compare the brightness of stars, allowing us to explore the vastness of space.
Graphing Logarithmic Functions
Graphing logarithmic functions is like drawing a picture of the pattern they create. They usually look like tilted S-shaped curves, where the x-axis is stretched out and the y-axis is squished.
Modeling with Logarithmic Functions
Finally, logarithmic functions are like magic wands that can transform real-world phenomena into mathematical models. They can represent population growth, bacterial decay, and even the decline of radioactive elements.
Creating a Logarithmic Model:
- Identify the pattern: Observe the exponential growth or decay and identify the percentage change.
- Choose a base: Select a base that makes the calculation easy.
- Write the equation: Express the model as
y = a * b^x, whereais the initial value,bis the base, andxis the exponent representing the percentage change.
Interpreting Logarithmic Models:
- Growth: The model predicts an exponential increase over time.
- Decay: The model predicts an exponential decrease over time.
- Half-life: The exponent represents the half-life, or the time it takes for the quantity to decrease or increase by half.
