Mathematics | Exponential and logarithms

Pure Mathematics 3: Comprehensive Study Notes on Exponential and Logarithms

Hello future mathematicians! Welcome to the exciting world of Exponential and Logarithmic functions. This chapter is absolutely central to P3 and forms the bedrock for advanced calculus (differentiation and integration, which come next!).

Don't worry if these functions look intimidating; they are simply tools we use to describe growth, decay, and change in the real world—from population dynamics and calculating loan interest to radioactive decay. By the end of these notes, you’ll be a pro at solving equations involving the special number \(e\).

1. Revisiting Exponential Functions \(y = a^x\)

An exponential function is characterized by having the variable \(x\) in the exponent (or power).

• Definition: The function \(f(x) = a^x\), where \(a\) is a constant known as the base and \(a > 0\).
• Graphs: All exponential graphs of the form \(y = a^x\) pass through the point \((0, 1)\) because \(a^0 = 1\) (as long as \(a \neq 0\)).
• Asymptotes: The \(x\)-axis (\(y=0\)) is a horizontal asymptote. The graph never touches or crosses the \(x\)-axis, meaning the output \(y\) is always positive.

Analogy: Think about exponential growth like a chain reaction. If you double your money every day (\(a=2\)), the growth starts slow (\(2^1=2\), \(2^2=4\)) but quickly becomes massive (\(2^{10}=1024\)).

Key Takeaway:

Exponential functions show growth or decay where the rate of change is proportional to the current amount. They are always positive and pass through \((0, 1)\).

2. Introducing the Natural Exponential Function \(y = e^x\)

In Pure Mathematics 3, we don't just use any base \(a\); we focus heavily on the most important base for describing natural processes: the number \(e\).

What is \(e\)?

The number \(e\) is an irrational number, often called Euler’s Number. Its value is approximately:

\(e \approx 2.71828\)

Why is \(e\) special?

The function \(y = e^x\) models continuous growth—growth where the rate of increase is exactly equal to the amount present at any given time. This makes it crucial for calculus.

• The function \(y = e^x\) is called the Natural Exponential Function.
• Its graph looks just like any other exponential graph \(y = a^x\), but its gradient at any point \((x, y)\) is simply \(y\). (We will explore this fully in the differentiation chapter!)

Did you know? \(e\) arises naturally when calculating continuously compounded interest. If you invest $1 at 100% interest compounded continuously for one year, you end up with exactly \(\$e\).

Key Takeaway:

\(e\) is the fundamental constant for natural growth and decay. In P3, \(e^x\) is your standard exponential function.

3. The Natural Logarithm \(\ln x\)

Logarithms are simply the inverse of exponential functions. They answer the question: "What power do I need to raise the base to, to get a certain number?"

The relationship between exponentials and logarithms is:
If \(y = a^x\), then \(x = \log_a y\).

In P3, because we use the base \(e\) so frequently, we use a special notation for the logarithm with base \(e\): the Natural Logarithm.

• Definition: The natural logarithm of \(x\) is written as \(\ln x\).
• Relationship to \(e\): \(\ln x\) means \(\log_e x\).
• Inverse Property: If \(y = e^x\), then \(x = \ln y\).

This inverse relationship is the most important concept in the chapter:

$$e^{\ln x} = x \quad \text{and} \quad \ln(e^x) = x$$

Analogy: If \(e^x\) is putting your gloves on, \(\ln x\) is taking them off. They are opposite operations that cancel each other out.

Graphing \(y = \ln x\)

Since \(y = \ln x\) is the inverse of \(y = e^x\), its graph is the reflection of \(y = e^x\) across the line \(y = x\).

• It passes through \((1, 0)\) because \(\ln(1) = 0\).
• The vertical axis (\(x=0\)) is a vertical asymptote. This means you can only take the logarithm of a positive number (i.e., the domain is \(x > 0\)).

Common Mistake to Avoid:
You cannot find \(\ln(0)\) or \(\ln\) of any negative number!

Key Takeaway:

\(\ln x\) is the operation that "undoes" \(e^x\). Remember that the domain of \(\ln x\) is strictly positive \(x\).

4. The Laws of Logarithms (Crucial Review)

To solve complex equations, you must master the three laws of logarithms. These laws apply regardless of the base, but in P3, we usually apply them to \(\ln\).

Let \(A\) and \(B\) be positive numbers and \(k\) be any real number.

Law 1: The Multiplication Law (Product Rule)

When you multiply terms inside the logarithm, you add the separate logarithms.

$$\ln (AB) = \ln A + \ln B$$

Law 2: The Division Law (Quotient Rule)

When you divide terms inside the logarithm, you subtract the separate logarithms.

$$\ln \left( \frac{A}{B} \right) = \ln A - \ln B$$

Law 3: The Power Law (Power Rule)

The exponent inside the logarithm can be moved to the front as a multiplier.

$$\ln (A^k) = k \ln A$$

Memory Aid: Think of the Power Law as "dropping" the power to the front. This is the single most important law for solving exponential equations!

Special Logarithm Results

These follow directly from the definition and are essential for simplification:

1. \(\ln (e) = 1\) (Because \(e^1 = e\)).
2. \(\ln (1) = 0\) (Because \(e^0 = 1\)).

5. Solving Exponential and Logarithmic Equations

Solving equations in P3 usually involves using \(\ln\) to "bring down" the unknown variable from the exponent.

Case A: Solving Exponential Equations (finding \(x\))

Goal: Isolate the exponential term, then apply \(\ln\) to both sides.

Example 1: Solving using base \(e\)
Solve \(4e^{2x} - 7 = 13\). (Give your answer to 3 significant figures.)

1. Isolate \(e\): Add 7 and divide by 4.
   $$4e^{2x} = 20 \implies e^{2x} = 5$$
2. Take \(\ln\) of both sides: This cancels the \(e\).
   $$\ln (e^{2x}) = \ln 5 \implies 2x = \ln 5$$
3. Solve for \(x\):
   $$x = \frac{\ln 5}{2}$$
4. Calculate: \(x \approx 0.805\) (3 s.f.)

Example 2: Solving equations with a base other than \(e\)
Solve \(3^{x+1} = 50\).

1. Take \(\ln\) of both sides: (We use \(\ln\) because it’s the standard P3 tool, even though we could use \(\log_{10}\) or \(\log_3\)).
   $$\ln (3^{x+1}) = \ln 50$$
2. Apply the Power Law: Drop the exponent to the front.
   $$(x+1) \ln 3 = \ln 50$$
3. Isolate \(x\): Divide by \(\ln 3\).
   $$x+1 = \frac{\ln 50}{\ln 3}$$
4. Final calculation:
   $$x = \frac{\ln 50}{\ln 3} - 1 \approx 3.5609 - 1 = 2.56$$ (3 s.f.)

Remember the golden rule: Always isolate the term containing \(x\) *before* applying \(\ln\).

Case B: Solving Logarithmic Equations (finding \(x\))

Goal: Combine log terms, then use the inverse operation \(e\).

Example 3: Solve \(\ln(x+2) - \ln(x-3) = 1\).

1. Combine using the Division Law:
   $$\ln \left( \frac{x+2}{x-3} \right) = 1$$
2. Use the inverse \(e\) (raise both sides as a power of \(e\)):
   $$e^{\ln \left( \frac{x+2}{x-3} \right)} = e^1$$
   $$\frac{x+2}{x-3} = e$$
3. Solve for \(x\):
   $$x+2 = e(x-3)$$
   $$x+2 = ex - 3e$$
   $$2 + 3e = ex - x$$
   $$2 + 3e = x(e - 1)$$
4. Final solution:
   $$x = \frac{2 + 3e}{e - 1} \approx 5.86$$ (3 s.f.)

Important Check: After solving a logarithmic equation, always check that your solution does not require taking the \(\ln\) of a negative number. Here, if \(x \approx 5.86\), both \((x+2)\) and \((x-3)\) are positive, so the solution is valid.

6. Modeling Real-World Data (Linearization)

A key skill in P3 is taking real-world data that follows an exponential relationship and transforming it into a linear relationship, \(Y = mX + c\), so that we can easily find the constants involved using a graph.

Don't worry if this seems tricky at first—the method is procedural and always the same: take \(\ln\) of both sides!

Model 1: Exponential Growth/Decay (\(y = Ab^x\))

This model describes how a quantity \(y\) changes over time \(x\) (e.g., population growth).

1. Take \(\ln\) of both sides:
   $$\ln y = \ln (Ab^x)$$
2. Use the Multiplication Law:
   $$\ln y = \ln A + \ln b^x$$
3. Use the Power Law:
   $$\ln y = \ln A + x (\ln b)$$

Now, compare this result to the linear form \(Y = c + m X\):

• Plot Y vs X: Plot \(\ln y\) (on the vertical axis, \(Y\)) against \(x\) (on the horizontal axis, \(X\)).
• Gradient \(m\): The gradient of the line is \(m = \ln b\). (To find \(b\), calculate \(b = e^m\)).
• Y-intercept \(c\): The intercept is \(c = \ln A\). (To find \(A\), calculate \(A = e^c\)).

Model 2: Power Laws (\(y = Ax^n\))

This model describes relationships where one quantity depends on a power of another (e.g., geometric scaling).

1. Take \(\ln\) of both sides:
   $$\ln y = \ln (Ax^n)$$
2. Use the Multiplication Law:
   $$\ln y = \ln A + \ln x^n$$
3. Use the Power Law:
   $$\ln y = \ln A + n (\ln x)$$

Compare this to \(Y = c + m X\):

• Plot Y vs X: Plot \(\ln y\) (on the vertical axis, \(Y\)) against \(\ln x\) (on the horizontal axis, \(X\)).
• Gradient \(m\): The gradient of the line is \(m = n\). (This gives the power directly.)
• Y-intercept \(c\): The intercept is \(c = \ln A\). (To find \(A\), calculate \(A = e^c\)).

Step-by-Step for Solving Modeling Problems:

1. Identify the given model (Is it \(y = Ab^x\) or \(y = Ax^n\)?).
2. Apply the logarithm (usually \(\ln\)) to linearize the equation.
3. Calculate the required transformed data points (e.g., find \(\ln y\) for all your \(y\) values).
4. Plot the transformed graph (e.g., \(\ln y\) vs \(x\)).
5. Calculate the gradient \(m\) and the intercept \(c\) from the straight line.
6. Use \(m\) and \(c\) to find the original constants \(A, b,\) or \(n\).

Key Takeaway:

Transforming exponential or power data involves taking the natural logarithm (\(\ln\)) of both sides to produce an equation of the form \(Y = mX + c\), allowing you to use linear graph properties (gradient and intercept) to find the unknown parameters.