Introduction: The Magic of "e"
Welcome to one of the most exciting chapters in A Level Maths! So far, you have worked with powers like \( x^2 \) or \( 10^x \). But in this chapter, we meet a very special number called e (Euler’s number), which is approximately 2.718.
Why is it special? Because it is the "language of nature." From the way bacteria grow in a petri dish to how your hot chocolate cools down, the number e is working behind the scenes. Don’t worry if it feels a bit "alien" at first—by the end of these notes, you'll see that e and its partner, the natural logarithm (ln), are just tools to help us solve real-world puzzles.
1. The Exponential Function \( y = e^x \)
The function \( y = e^x \) is a specific type of exponential function where the base is the constant e.
The Shape of the Graph
The graph of \( y = e^x \) has some very specific features you need to know:
• It always stays above the x-axis (the y-values are always positive).
• It passes through the point (0, 1) because any number to the power of 0 is 1.
• As \( x \) gets very large, the graph shoots up incredibly fast (exponential growth).
• As \( x \) gets very negative, the graph gets closer and closer to the x-axis but never touches it. We call the x-axis a horizontal asymptote.
The "Special Power" of \( e^x \)
In your syllabus (Ref: E9), there is a unique rule: the gradient (steepness) of the curve \( y = e^x \) at any point is exactly equal to the y-value at that point!
If \( y = e^x \), then the gradient \( \frac{dy}{dx} = e^x \).
Differentiating \( e^{kx} \)
If there is a number (\( k \)) in front of the \( x \), the rule changes slightly:
If \( y = e^{kx} \), then the gradient is \( \frac{dy}{dx} = ke^{kx} \).
Example: If \( y = e^{5x} \), then the gradient is \( 5e^{5x} \).
Quick Review Box:
• e ≈ 2.718
• \( e^0 = 1 \)
• The gradient of \( e^{kx} \) is \( ke^{kx} \).
2. The Natural Logarithm: \( y = \ln x \)
Think of the natural logarithm (written as ln) as the "undo button" for e. If you have an equation with \( e^x \) and you want to find \( x \), you use ln. (Ref: E10)
Key Facts about ln:
• ln x is actually just \( \log_e x \). It is a logarithm with base e.
• The Inverse Relationship: \( y = e^x \) and \( y = \ln x \) are inverse functions. This means they reflect each other across the diagonal line \( y = x \).
• Cancellation: Because they are inverses, \( \ln(e^x) = x \) and \( e^{\ln x} = x \).
The Shape of the Graph
The graph of \( y = \ln x \) is the reflection of \( y = e^x \):
• It only exists for positive values of x (you cannot take the log of 0 or a negative number!).
• It passes through (1, 0) because \( \ln(1) = 0 \).
• The y-axis is a vertical asymptote (the graph gets very close but never touches it).
Did you know?
The "ln" stands for logarithme naturel (French for natural logarithm). Most people just pronounce it "ell-enn."
Key Takeaway: If you want to move an \( x \) down from a power in an \( e^x \) equation, just "take ln" of both sides!
3. Solving Equations with \( e \) and \( \ln \)
To solve these, we use the fact that they "cancel" each other out. (Ref: E6)
Step-by-Step: Solving for \( x \)
Example A: Solving \( e^{2x} = 10 \)
1. Take the natural log of both sides: \( \ln(e^{2x}) = \ln(10) \)
2. Use the "undo" rule: \( 2x = \ln(10) \)
3. Divide by 2: \( x = \frac{\ln(10)}{2} \)
4. Use your calculator: \( x \approx 1.15 \)
Example B: Solving \( \ln(x + 1) = 4 \)
1. "E" both sides (put both sides as powers of e): \( e^{\ln(x+1)} = e^4 \)
2. Use the "undo" rule: \( x + 1 = e^4 \)
3. Subtract 1: \( x = e^4 - 1 \)
4. Use your calculator: \( x \approx 53.6 \)
Common Mistakes to Avoid:
• Don't try to take ln of a negative number: If your calculation results in \( \ln(-5) \), you've likely made a sign error!
• The coefficient must be 1: Before taking ln of both sides, make sure the \( e^x \) part is on its own. If you have \( 3e^x = 12 \), divide by 3 first to get \( e^x = 4 \).
4. Exponential Growth and Decay Modelling
We use e to model things that change at a rate proportional to their current size (Ref: E11). The standard formula is:
\( N = N_0 e^{kt} \)
What do the letters mean?
• \( N \): The amount at time \( t \).
• \( N_0 \): The initial amount (what you started with at \( t = 0 \)).
• \( k \): The growth constant. If \( k \) is positive, it’s growth (like a population). If \( k \) is negative, it’s decay (like radioactive waste or cooling tea).
• \( t \): Time.
Analogy: The Compound Interest Bank Account
Imagine a bank account that pays interest every single second of every day. Instead of growing in "steps" every year, it grows in a smooth, elegant curve. That smooth curve is exactly what \( e^x \) represents—continuous growth.
Steps for Modelling Questions:
1. Find \( N_0 \): Look for the "starting value" in the question.
2. Find \( k \): Use a known set of values (e.g., "after 5 years, the population was 200") to solve for \( k \).
3. Answer the puzzle: Once you have the full formula, you can find \( N \) for any time \( t \), or find \( t \) for any amount \( N \).
Key Takeaway: Exponential models are great for the short term, but always consider limitations. For example, a population can't grow forever because it will eventually run out of food or space!
Summary Checklist
Can you...
• Sketch the graphs of \( y = e^x \) and \( y = \ln x \)?
• Remember that \( \frac{dy}{dx} (e^{kx}) = ke^{kx} \)?
• Use ln to solve equations where \( x \) is a power of e?
• Identify the initial value \( N_0 \) in a growth/decay word problem?
• Explain why \( \ln x \) only exists for \( x > 0 \)?
Don't worry if this seems tricky at first! These functions are just a new way of describing patterns we see in the world every day. Keep practicing those "undo" steps, and it will soon become second nature!