M1 Study Notes: Exponential and Logarithmic Functions
Hey there! Welcome to one of the most powerful topics in M1 – Exponential and Logarithmic Functions. Don't worry if the names sound a bit intimidating. These are just special types of functions that are amazing at describing things that grow or shrink very, very quickly.
Think about how a viral video spreads, how your money can grow in a bank account, or how scientists measure the age of ancient fossils. All of these real-world situations use the ideas we're about to learn. In this chapter, we'll explore the magical number 'e', understand its function $$y = e^x$$, meet its inverse partner $$y = \ln x$$, and learn how to use them to solve practical problems.
Let's take it one step at a time. You've got this!
1. The Magical Number 'e'
You've heard of π (pi), right? Well, 'e' is another super important mathematical constant. It's an irrational number, which means its decimals go on forever without repeating.
Value of e: $$e \approx 2.71828...$$
But where does it come from? It's formally defined by an infinite series. This might look complex, but it's just a special recipe for calculating 'e'.
The exponential series for $$e^x$$ is:
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...$$(Remember, $$n!$$ means "n factorial", e.g., $$3! = 3 \times 2 \times 1 = 6$$)
To find the value of 'e' itself, we just plug in $$x=1$$ into this series:
$$e^1 = e = 1 + 1 + \frac{1^2}{2!} + \frac{1^3}{3!} + \frac{1^4}{4!} + ... = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + ... \approx 2.71828$$Did you know?
The number 'e' is all about continuous growth. It appears naturally in finance when calculating continuously compounded interest, in biology for population growth models, and in physics for radioactive decay. It's sometimes called Euler's number, after the Swiss mathematician Leonhard Euler.
Key Takeaway
'e' is a special constant approximately equal to 2.718. It's the base for all things related to natural growth and decay, and it's defined by the exponential series.
2. The Natural Exponential Function: $$y = e^x$$
This is the star of the show! The function $$f(x) = e^x$$ is the "base" function for modeling rapid growth.
Graph and Key Features of $$y = e^x$$:
- Passes through (0, 1): Anything to the power of 0 is 1, so $$e^0 = 1$$.
- Always Positive: The graph is always above the x-axis. You can never raise 'e' to a power and get a negative number or zero.
- Horizontal Asymptote: As $$x$$ becomes very negative (e.g., -100, -1000), $$e^x$$ gets incredibly close to 0, but never touches it. We say the x-axis (the line $$y=0$$) is a horizontal asymptote.
- Rapid Growth: As $$x$$ increases, the value of $$y$$ grows extremely fast. This is what "exponential growth" means!
Analogy: Think of a snowball rolling down a hill. It starts small, but as it rolls, it picks up more snow faster and faster. That's exponential growth!
Key Takeaway
The function $$y=e^x$$ models natural growth. Its graph starts near zero, passes through the point (0, 1), and then shoots upwards very quickly.
3. The Natural Logarithmic Function: $$y = \ln x$$
Every hero has a partner, and for $$e^x$$, that partner is the natural logarithm, written as $$\ln x$$.
What does $$\ln x$$ mean? It's the "undo" button for $$e^x$$.
$$\ln(a)$$ asks the question: "What power do I need to raise 'e' to, in order to get 'a'?"
Examples:
- $$\ln(e) = 1$$, because $$e^1 = e$$.
- $$\ln(1) = 0$$, because $$e^0 = 1$$.
- $$\ln(e^5) = 5$$, because the power you need to raise 'e' to is 5.
The functions $$y = e^x$$ and $$y = \ln x$$ are inverse functions. This means they cancel each other out.
$$ \ln(e^x) = x $$ $$ e^{\ln x} = x $$Graph and Key Features of $$y = \ln x$$:
- Passes through (1, 0): As we saw, $$\ln(1) = 0$$.
- Only for positive x: You cannot take the logarithm of a negative number or zero. The domain is $$x > 0$$.
- Vertical Asymptote: As $$x$$ gets very close to 0 from the positive side, $$\ln x$$ becomes a very large negative number. The y-axis (the line $$x=0$$) is a vertical asymptote.
- Slow Growth: The graph increases, but much more slowly than the exponential function.
- Reflection: The graph of $$y=\ln x$$ is a perfect reflection of the graph of $$y=e^x$$ across the line $$y=x$$.
Key Takeaway
The function $$y=\ln x$$ is the inverse of $$y=e^x$$. It's only defined for positive x, passes through (1, 0), and grows slowly.
4. Solving Exponential and Logarithmic Equations
To solve problems, we need to be comfortable with the rules and how to rearrange equations. Don't worry, it's all about using the inverse relationship and some basic log laws.
Quick Review: Laws of Logarithms
These work for any base, but we'll use them with 'ln'.
- Product Rule: $$\ln(ab) = \ln(a) + \ln(b)$$
- Quotient Rule: $$\ln(\frac{a}{b}) = \ln(a) - \ln(b)$$
- Power Rule: $$\ln(a^n) = n \ln(a)$$
Step-by-Step Guide to Solving Equations:
Case 1: Solving Exponential Equations (where the unknown is in the power)
Example: Solve for x in $$e^{2x} = 5$$
- Isolate the exponential term. (It's already done here).
- Take the natural log (ln) of both sides. This is the key step to "bring the power down".
$$\ln(e^{2x}) = \ln(5)$$ - Use the inverse property $$\ln(e^{\text{stuff}}) = \text{stuff}$$.
$$2x = \ln(5)$$ - Solve for x.
$$x = \frac{\ln(5)}{2} \approx \frac{1.6094}{2} \approx 0.805$$
Case 2: Solving Logarithmic Equations (where the unknown is inside a log)
Example: Solve for x in $$\ln(x-3) = 2$$
- Isolate the log term. (Already done).
- "Exponentiate" both sides (make both sides the power of 'e'). This is the "undo" step for ln.
$$e^{\ln(x-3)} = e^2$$ - Use the inverse property $$e^{\ln(\text{stuff})} = \text{stuff}$$.
$$x-3 = e^2$$ - Solve for x.
$$x = e^2 + 3 \approx 7.389 + 3 \approx 10.389$$
Common Mistake Alert!
Always remember that you can only take the log of a positive number. After solving a log equation, you should check your answer by plugging it back into the original equation to make sure the term inside the 'ln' is positive. In the example above, $$x \approx 10.389$$, so $$x-3 \approx 7.389$$, which is positive. So the solution is valid!
5. Real-World Applications
This is where it all comes together! These functions are used to model many real-life situations.
A. Population Growth
Unrestricted populations often grow exponentially.
Formula: $$P(t) = P_0 e^{kt}$$
- $$P(t)$$ is the population at time $$t$$.
- $$P_0$$ is the initial population (at $$t=0$$).
- $$k$$ is the relative growth rate (a positive constant).
- $$t$$ is time.
Example: A bacterial culture starts with 500 bacteria. After 3 hours, there are 8000 bacteria. Find the number of bacteria after 4 hours.
First, find k. We have $$P_0=500$$, $$t=3$$, $$P(3)=8000$$.
$$8000 = 500 e^{k(3)}$$
$$16 = e^{3k}$$
$$\ln(16) = \ln(e^{3k})$$
$$\ln(16) = 3k \implies k = \frac{\ln(16)}{3} \approx 0.924$$
Now, find P(4):
$$P(4) = 500 e^{0.924 \times 4} \approx 500 e^{3.696} \approx 20153$$ bacteria.
B. Radioactive Decay
Radioactive substances decay exponentially over time.
Formula: $$A(t) = A_0 e^{-kt}$$
- $$A(t)$$ is the amount remaining at time $$t$$.
- $$A_0$$ is the initial amount.
- $$k$$ is the decay constant (a positive constant). Notice the negative sign in the exponent, which causes the decay!
- $$t$$ is time.
C. Continuously Compounded Interest
The ultimate goal for savings! This is when interest is calculated and added an infinite number of times per year.
Formula: $$A = P e^{rt}$$
- $$A$$ is the final amount of money.
- $$P$$ is the principal (initial amount).
- $$r$$ is the annual interest rate (as a decimal).
- $$t$$ is the number of years.
6. Linearization: Turning Curves into Straight Lines
Sometimes in experiments, we collect data that looks like it follows an exponential curve, like $$y=ka^x$$. It's hard to be sure just by looking at a curve. It's much easier to tell if points lie on a straight line!
Linearization is a clever trick using logarithms to transform a curve into a straight line. The equation of a straight line is $$Y = mX + C$$, where $$m$$ is the slope and $$C$$ is the Y-intercept.
Case 1: Transforming $$y = ka^x$$
This model is common in biology and finance.
- Start with the equation: $$y = ka^x$$
- Take the natural log of both sides: $$\ln(y) = \ln(ka^x)$$
- Use the product rule for logs: $$\ln(y) = \ln(k) + \ln(a^x)$$
- Use the power rule for logs: $$\ln(y) = \ln(k) + x \ln(a)$$
- Rearrange to match $$Y = mX + C$$:
$$\ln(y) = (\ln a)x + \ln(k)$$
Now, compare this to $$Y=mX+C$$:
- The new Y-axis is $$\ln(y)$$.
- The new X-axis is just $$x$$.
- The slope (m) is $$\ln(a)$$.
- The Y-intercept (C) is $$\ln(k)$$.
So, if you plot a graph of $$\ln(y)$$ against $$x$$, you should get a straight line! From the graph, you can find the slope and intercept, and use them to calculate the original constants, $$a$$ and $$k$$.
$$a = e^{\text{slope}}$$ and $$k = e^{\text{y-intercept}}$$
Case 2: Transforming $$y = k[f(x)]^n$$ (e.g., $$y = kx^n$$)
This is a power law model, common in physics and engineering.
- Start with the equation: $$y = kx^n$$ (here, $$f(x)=x$$)
- Take the natural log of both sides: $$\ln(y) = \ln(kx^n)$$
- Use the product rule: $$\ln(y) = \ln(k) + \ln(x^n)$$
- Use the power rule:
$$\ln(y) = n \ln(x) + \ln(k)$$
Compare this to $$Y=mX+C$$:
- The new Y-axis is $$\ln(y)$$.
- The new X-axis is $$\ln(x)$$.
- The slope (m) is $$n$$.
- The Y-intercept (C) is $$\ln(k)$$.
So, if you plot a graph of $$\ln(y)$$ against $$\ln(x)$$, you will get a straight line. The slope of this line is your constant $$n$$, and from the y-intercept, you can find $$k$$ using $$k = e^{\text{y-intercept}}$$.
Key Takeaway
Linearization uses logarithms to transform exponential or power-law relationships into straight lines. By plotting the correct variables (like $$\ln y$$ vs $$x$$), we can find the unknown constants from the slope and y-intercept of the resulting line.
And that's a wrap on exponential and logarithmic functions! We've journeyed from the definition of 'e' to using its powers to model the world and analyze data. Keep practising the solving and linearization techniques, and you'll find these functions are your friends, not your foes. Good luck!