Exponentials and logarithms

Welcome to Exponentials and Logarithms!

In this chapter, we are going to explore some of the most powerful tools in mathematics. Exponentials and logarithms are used everywhere—from tracking how a virus spreads and calculating compound interest in your bank account, to measuring the loudness of sound or the intensity of earthquakes. Don't worry if these seem a bit "alien" at first; once you see the patterns, they become much easier to handle! We will break everything down step-by-step.

1. Exponential Functions and the Number \( e \)

An exponential function is one where the variable \( x \) is in the "power" position, like \( a^x \). The number \( a \) is called the base.

The Shape of the Graph

The shape of the graph \( y = a^x \) depends on the value of the base \( a \):
• If \( a > 1 \): The graph shoots upwards. This is called exponential growth. Imagine a bank account where your money doubles every year!
• If \( 0 < a < 1 \): The graph slides downwards towards the x-axis. This is called exponential decay. Imagine the value of a car dropping over time.

Important Point: All graphs of the form \( y = a^x \) pass through the point (0, 1) because any number to the power of zero is 1. They also never actually touch the x-axis; the x-axis is an asymptote.

The Natural Number \( e \)

In A Level Maths, we use a very special base called \( e \) (approximately 2.718). It is often called Euler's number. Why is it special? Because for the function \( y = e^x \), the gradient (the slope) at any point is exactly the same as the y-value at that point!

Key Rule for Differentiation:
If \( y = e^{kx} \), then the gradient is \( \frac{dy}{dx} = ke^{kx} \).
Example: If \( y = e^{3x} \), the derivative is \( 3e^{3x} \).

Key Takeaway: Exponential functions grow or decay at a rate proportional to their current size. The number \( e \) is the "magic" base where the growth rate equals the value itself.

2. Introduction to Logarithms

If exponentials are like "squaring" a number, logarithms are like finding the "square root." They are inverses. A logarithm tells you what power you need to raise a base to in order to get a certain number.

The Definition:
If \( a^y = x \), then \( \log_a x = y \).

Memory Aid: "The base stays the base."
In \( a^y = x \), the base is \( a \). In \( \log_a x = y \), the base is still \( a \) (the little number at the bottom). The other two numbers just swap sides!

Natural Logarithms (\( \ln x \))

Just like \( e \) is our special exponential base, we have a special logarithm base. A logarithm with base \( e \) is written as \( \ln x \) (pronounced "len x").
• \( \ln x \) is the inverse of \( e^x \).
• This means \( \ln(e^x) = x \) and \( e^{\ln x} = x \). They "cancel" each other out.

Did you know? The graph of \( y = \ln x \) is just the graph of \( y = e^x \) reflected in the line \( y = x \). Because you can't have a power that results in a negative number, you can only take the log of a positive number (\( x > 0 \)).

3. The Laws of Logarithms

To solve tricky equations, you need to know the three main "Laws of Logs." These work for any base (as long as the base is the same for all terms in the equation).

1. The Multiplication Law: \( \log_a x + \log_a y = \log_a (xy) \)
(Adding logs is like multiplying the numbers inside.)

2. The Division Law: \( \log_a x - \log_a y = \log_a (\frac{x}{y}) \)
(Subtracting logs is like dividing the numbers inside.)

3. The Power Law: \( \log_a (x^k) = k \log_a x \)
(You can move a power to the front of the log like a multiplier.)

Quick Review Box:
• \( \log_a a = 1 \) (Because \( a^1 = a \))
• \( \log_a 1 = 0 \) (Because \( a^0 = 1 \))
• Common Mistake: \( \log(x+y) \) is NOT \( \log x + \log y \). The laws only work when you add/subtract the logs themselves!

4. Solving Exponential Equations

Sometimes you need to find \( x \) when it is stuck in the power, like \( 3^x = 20 \). We use logs to "bring the \( x \) down."

Step-by-Step Process:
1. Take logs of both sides: \( \log(3^x) = \log(20) \)
2. Use the Power Law to bring \( x \) to the front: \( x \log 3 = \log 20 \)
3. Divide to find \( x \): \( x = \frac{\log 20}{\log 3} \)
4. Use your calculator to get the decimal answer.

Don't worry if this seems tricky at first! Just remember that logarithms were practically invented to solve for unknown powers.

5. Using Logarithms for Graphs (Linearization)

Scientists often collect data that follows an exponential curve, but curves are hard to read. We use logarithms to turn these curves into straight lines (\( y = mx + c \)).

Type 1: \( y = ax^n \) (The Power Law)

If we take logs of both sides: \( \log y = \log(ax^n) \)
Using our laws: \( \log y = n \log x + \log a \)
This looks like \( Y = mX + c \), where:
• We plot \( \log y \) against \( \log x \).
• The gradient is \( n \).
• The intercept is \( \log a \).

Type 2: \( y = kb^x \) (The Exponential Law)

If we take logs of both sides: \( \log y = \log(kb^x) \)
Using our laws: \( \log y = (\log b)x + \log k \)
This looks like \( Y = mX + c \), where:
• We plot \( \log y \) against \( x \).
• The gradient is \( \log b \).
• The intercept is \( \log k \).

Key Takeaway: If you see a graph with \( \log \) on both axes, it's Type 1. If you see \( \log \) on only the vertical axis, it's Type 2!

6. Modelling Growth and Decay

In the real world, we use the formula \( V = Ae^{kt} \) to model things.
• \( V \) is the value at time \( t \).
• \( A \) is the initial value (the value when \( t = 0 \)).
• \( k \) is the growth constant. If \( k \) is positive, it's growth; if \( k \) is negative, it's decay.

Analogy: Think of \( A \) as your starting line in a race, and \( k \) as your speed. The further you go (\( t \)), the faster you pull away from the start!

Limitations of Models

Always check if your answer makes sense. For example, a population model might predict a billion people in a small village after 100 years. In reality, the model is limited by space, food, or other factors. Always comment on the context if the question asks for limitations!

Summary of Modelling:
• "Initial" means set \( t = 0 \).
• "Rate of change" means find the gradient (\( \frac{dV}{dt} \)).
• To find when something reaches a certain level, substitute that value for \( V \) and solve for \( t \) using logs.