Exponentials and Logarithms: CCEA AS 1 Pure Mathematics Study Notes
Welcome to one of the most powerful and exciting topics in AS 1 Pure Mathematics! Whether you are tracking the spread of a virus, calculating how investments grow, or measuring radioactive decay, exponentials and logarithms are the mathematical tools that make it possible. Don't worry if this topic feels unfamiliar at first — we will break down every concept step-by-step with clear examples, memory aids, and examiner tips so you can tackle exam questions with total confidence.
Unit context: This chapter is part of Unit AS 1: Pure Mathematics (SMT11) for the CCEA GCE Mathematics specification (Subject Code: 2210).
---1. The Exponential Function: \(y = a^x\) and \(y = \text{e}^x\)
Understanding \(y = a^x\)
An exponential function is a function where the variable appears in the power (exponent), such as \(y = 2^x\) or \(y = 10^x\). The standard form is \(y = a^x\), where \(a > 0\) and \(a \neq 1\).
Key features of the graph \(y = a^x\) (for \(a > 1\)):
• \(y\)-intercept: It always crosses the \(y\)-axis at \((0, 1)\) because any non-zero number to the power of zero equals \(1\) (\(a^0 = 1\)).
• Horizontal Asymptote: The line \(y = 0\) (the \(x\)-axis) is a horizontal asymptote. As \(x \to -\infty\), \(y \to 0\), but the curve never actually touches or crosses the \(x\)-axis.
• Domain and Range: The domain is all real numbers (\(x \in \mathbb{R}\)), and the range is strictly positive values (\(y > 0\)).
The Special Number \(\text{e}\) and the Natural Exponential Function
In mathematics, there is a special irrational number denoted by \(\text{e}\), known as Euler's number, where \(\text{e} \approx 2.71828\). Just like \(\pi\), it appears naturally throughout science and finance.
The function \(y = \text{e}^x\) is called the natural exponential function. It has a unique and beautiful property:
The gradient (rate of change) of the curve at any point is exactly equal to the \(y\)-value at that point!
In calculus terms: \(\frac{\mathrm{d}}{\mathrm{d}x}(\text{e}^x) = \text{e}^x\). More generally, for a linear power: \(\frac{\mathrm{d}}{\mathrm{d}x}(\text{e}^{kx}) = k\text{e}^{kx}\).
Key Takeaway: All graphs of \(y = a^x\) pass through \((0, 1)\) and approach the \(x\)-axis as a horizontal asymptote. The natural exponential base \(\text{e} \approx 2.71828\) is unique because its rate of growth equals its value.
---2. Logarithms: The Inverse of Exponentials
What is a Logarithm?
Think of a logarithm as an undo button for powers. A logarithm answers the question: "To what power must I raise the base to get this number?"
The Fundamental Equivalence:
\(y = a^x \iff x = \log_a y\) (where \(a > 0\), \(a \neq 1\), and \(y > 0\))
Example: Since \(2^3 = 8\), it follows that \(\log_2 8 = 3\).
Example: Since \(10^2 = 100\), it follows that \(\log_{10} 100 = 2\).
The Natural Logarithm: \(\ln x\)
When the base of the logarithm is \(\text{e}\), it is called the natural logarithm, written as \(\ln x\) instead of \(\log_{\text{e}} x\):
\(y = \text{e}^x \iff x = \ln y\)
Because \(\text{e}^x\) and \(\ln x\) are inverse functions of each other, they cancel each other out:
• \(\ln(\text{e}^x) = x\)
• \(\text{e}^{\ln x} = x\) (for \(x > 0\))
Graph of \(y = \ln x\):
• It is the reflection of \(y = \text{e}^x\) in the line \(y = x\).
• \(x\)-intercept: Passes through \((1, 0)\) because \(\ln(1) = 0\).
• Vertical Asymptote: The line \(x = 0\) (the \(y\)-axis). As \(x \to 0^+\), \(y \to -\infty\).
• Domain: \(x > 0\) (you cannot take the log of zero or a negative number).
• Range: All real numbers (\(y \in \mathbb{R}\)).
The 6 Essential Laws of Logarithms
These laws apply to logarithms of any valid base (including \(\ln\) and \(\log_{10}\)):
1. Multiplication Law: \(\log_a(xy) = \log_a x + \log_a y\)
2. Division Law: \(\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y\)
3. Power Law: \(\log_a(x^k) = k\log_a x\)
4. Log of 1: \(\log_a 1 = 0\)
5. Log of the Base: \(\log_a a = 1\) (and \(\ln \text{e} = 1\))
6. Reciprocal Law: \(\log_a\left(\frac{1}{x}\right) = -\log_a x\)
Key Takeaway: Logarithms and exponentials undo each other. Remember the golden rule: you can only evaluate \(\log_a x\) when \(x > 0\).
---3. Solving Exponential and Logarithmic Equations
Type 1: Solving \(a^x = b\) or \(a^{f(x)} = b\)
When the unknown is in the index, take logarithms (either \(\ln\) or \(\log_{10}\)) of both sides, then use the power law to bring the power down.
Step-by-Step Example: Solve \(3^{2x - 1} = 7\), giving your answer to 3 significant figures.
Step 1: Take \(\ln\) of both sides: \(\ln(3^{2x - 1}) = \ln 7\)
Step 2: Bring the power down using the power law: \((2x - 1)\ln 3 = \ln 7\)
Step 3: Divide by \(\ln 3\): \(2x - 1 = \frac{\ln 7}{\ln 3}\)
Step 4: Rearrange for \(x\): \(2x = \frac{\ln 7}{\ln 3} + 1 \implies x = \frac{1}{2}\left(\frac{\ln 7}{\ln 3} + 1\right)\)
Step 5: Calculate: \(x \approx 1.39\) (3 s.f.).
Type 2: Equations Reducible to Quadratics
Equations with terms like \(\text{e}^{2x}\) and \(\text{e}^x\), or \(a^{2x}\) and \(a^x\), can be disguised quadratic equations. Use the substitution \(u = \text{e}^x\) (or \(u = a^x\)), noting that \(\text{e}^{2x} = (\text{e}^x)^2 = u^2\).
Step-by-Step Example: Solve \(\text{e}^{2x} - 5\text{e}^x + 6 = 0\).
Step 1: Let \(u = \text{e}^x\). The equation becomes \(u^2 - 5u + 6 = 0\).
Step 2: Factorise: \((u - 2)(u - 3) = 0\), so \(u = 2\) or \(u = 3\).
Step 3: Substitute back \(u = \text{e}^x\):
• \(\text{e}^x = 2 \implies x = \ln 2 \approx 0.693\)
• \(\text{e}^x = 3 \implies x = \ln 3 \approx 1.10\)
Crucial Check: If one of the quadratic roots gives a negative value, e.g., \(u = -4 \implies \text{e}^x = -4\), you must reject this solution because \(\text{e}^x > 0\) for all real \(x\).
Type 3: Solving Logarithmic Equations
Combine multiple log terms into a single logarithm using the laws of logarithms before removing the log.
Step-by-Step Example: Solve \(\log_2(x + 6) - \log_2 x = 3\).
Step 1: Use the division law: \(\log_2\left(\frac{x + 6}{x}\right) = 3\)
Step 2: Rewrite in exponential form: \(\frac{x + 6}{x} = 2^3 = 8\)
Step 3: Solve for \(x\): \(x + 6 = 8x \implies 7x = 6 \implies x = \frac{6}{7}\)
Step 4: Check validity: Since \(\frac{6}{7} > 0\) and \(\frac{6}{7} + 6 > 0\), the solution is valid.
Key Takeaway: Always check your solutions in logarithmic equations! Any value that makes the argument inside a \(\log\) or \(\ln\) zero or negative is invalid and must be rejected.
---4. Reduction of Non-Linear Data to Linear Form
In scientific experiments, relationships between variables are often curves rather than straight lines. By taking logarithms, we can convert these non-linear relationships into the straight-line form \(Y = mX + c\), allowing us to find unknown constants from experimental graphs.
Model 1: Power Relationship \(y = a x^n\)
Take natural logarithms (or \(\log_{10}\)) of both sides:
\(\ln y = \ln(a x^n)\)
\(\ln y = \ln a + \ln(x^n)\)
\(\ln y = n \ln x + \ln a\)
Compare this to \(Y = mX + c\):
• Plotted Variables: Plot \(Y = \ln y\) against \(X = \ln x\)
• Gradient: \(m = n\)
• Vertical Intercept: \(c = \ln a \implies a = \text{e}^c\) (or \(a = 10^c\) if using \(\log_{10}\))
Model 2: Exponential Relationships
Case A: \(y = a b^x\)
Take \(\ln\) of both sides:
\(\ln y = \ln(a b^x) = \ln a + \ln(b^x)\)
\(\ln y = (\ln b)x + \ln a\)
• Plotted Variables: Plot \(Y = \ln y\) against \(X = x\)
• Gradient: \(m = \ln b \implies b = \text{e}^m\)
• Vertical Intercept: \(c = \ln a \implies a = \text{e}^c\)
Case B: \(y = A\text{e}^{kx}\)
Take \(\ln\) of both sides:
\(\ln y = \ln(A\text{e}^{kx}) = \ln A + \ln(\text{e}^{kx})\)
\(\ln y = kx + \ln A\)
• Plotted Variables: Plot \(Y = \ln y\) against \(X = x\)
• Gradient: \(m = k\)
• Vertical Intercept: \(c = \ln A \implies A = \text{e}^c\)
Quick Summary Table:
• If the graph is \(\ln y\) against \(\ln x\), the model is a power law \(y = ax^n\).
• If the graph is \(\ln y\) against \(x\), the model is an exponential relationship \(y = ab^x\) or \(y = A\text{e}^{kx}\).
Key Takeaway: Look carefully at the horizontal axis! If the axis is \(\ln x\), it is a power relationship (\(y = ax^n\)). If the axis is simply \(x\), it is an exponential relationship (\(y = ab^x\) or \(y = A\text{e}^{kx}\)).
---5. Exponential Growth and Decay Modelling
Many real-world quantities grow or decay at a rate proportional to their current size. These are modelled using:
\(N = N_0 \text{e}^{kt}\) (or \(N = N_0 a^t\))
Understanding the Parameters:
• \(N\): Quantity at time \(t\).
• \(N_0\): Initial value (the value when \(t = 0\), since \(\text{e}^0 = 1\)).
• \(k\): The growth/decay constant.
- If \(k > 0\), the model represents exponential growth (e.g., population growth, compound interest).
- If \(k < 0\), the model represents exponential decay (e.g., radioactive decay, drug clearance).
Finding Doubling Time and Half-Life
• Doubling Time: The time taken for the initial quantity to double (\(N = 2N_0\)).
\(2N_0 = N_0 \text{e}^{kt} \implies 2 = \text{e}^{kt} \implies t = \frac{\ln 2}{k}\)
• Half-Life: The time taken for the initial quantity to halve (\(N = \frac{1}{2}N_0\)).
\(\frac{1}{2}N_0 = N_0 \text{e}^{kt} \implies \frac{1}{2} = \text{e}^{kt} \implies \ln(0.5) = kt \implies t = \frac{\ln(0.5)}{k} = -\frac{\ln 2}{k}\)
Model Limitations
In examination questions, you may be asked to comment on the validity or limitations of an exponential model:
• Exponential growth models predict that \(N \to \infty\) as \(t \to \infty\). In reality, growth is eventually limited by physical constraints such as food supply, space, or resources.
• Exponential decay models predict that \(N \to 0\) as \(t \to \infty\), but never actually reaches zero.
Key Takeaway: At \(t = 0\), \(N = N_0\). To find the time for any target value, substitute the value for \(N\), isolate the exponential term, and take the natural logarithm of both sides.
---6. Common Pitfalls and Examiner Warnings
CCEA examiner reports frequently highlight the following common mistakes. Make sure to avoid them:
1. False Logarithm Distributive Rules (Very Common Error!):
• INCORRECT: \(\log(A + B) = \log A + \log B\) (Logs do not distribute over addition!)
• CORRECT: \(\log(AB) = \log A + \log B\)
• INCORRECT: \(\frac{\log A}{\log B} = \log A - \log B\)
• CORRECT: \(\log\left(\frac{A}{B}\right) = \log A - \log B\)
2. Linearisation Intercept Mistake:
When you find the vertical intercept \(c\) from a log-linear graph, remember that \(c = \ln a\). You must calculate \(a = \text{e}^c\) (or \(a = 10^c\)). Forgetting to exponentiate and writing \(a = c\) will lose marks!
3. Mixing Log Bases:
Do not mix \(\ln\) (base \(\text{e}\)) and \(\log_{10}\) within the same problem. If you take \(\ln\) of the left-hand side, you must take \(\ln\) of the right-hand side.
4. Failing to Reject Extraneous / Invalid Roots:
When solving logarithmic equations, always verify that your final solutions give positive arguments in the original log terms. If a solution gives \(\log(\text{negative number})\), it must be rejected.
5. Premature Rounding:
In multi-step growth and decay questions, do not round intermediate values of constants like \(k\) to 2 or 3 decimal places. Store exact values in your calculator memory. Rounding too early causes inaccuracies in your final answer (CCEA requires 3 significant figures unless specified otherwise).
Chapter Quick Review
• Equivalence: \(y = a^x \iff x = \log_a y\) and \(y = \text{e}^x \iff x = \ln y\)
• Key Laws: \(\ln(xy) = \ln x + \ln y\), \(\ln\left(\frac{x}{y}\right) = \ln x - \ln y\), \(\ln(x^k) = k\ln x\)
• Power Model: \(y = ax^n \implies \ln y = n\ln x + \ln a\) (Gradient \(= n\), Intercept \(= \ln a\))
• Exponential Model: \(y = ab^x \implies \ln y = (\ln b)x + \ln a\) (Gradient \(= \ln b\), Intercept \(= \ln a\))
• Natural Exponential Model: \(y = A\text{e}^{kx} \implies \ln y = kx + \ln A\) (Gradient \(= k\), Intercept \(= \ln A\))
• Modelling: \(N = N_0\text{e}^{kt}\), where \(N_0\) is the initial value at \(t = 0\).