📈 Exponential Functions and Equations
Basic introduction
🎯 Part A: The Exponential Function
Definition
An exponential function is a function in which the variable appears in the exponent (the power).
\(f(x) = a^x\)
where \(a > 0\) and \(a \neq 1\)
General form:
\(f(x) = b \cdot a^x + c\)
📊 Properties of \(f(x) = a^x\)
| Property | \(a > 1\) | \(0 < a < 1\) |
|---|---|---|
| Behaviour | 📈 Increasing | 📉 Decreasing |
| Domain | \(\mathbb{R}\) (all real numbers) | |
| Range | \((0, \infty)\) (positive only) | |
| Fixed point | \((0, 1)\) because \(a^0 = 1\) | |
| Asymptote | x-axis (the line \(y = 0\)) | |
| y-intercept | \((0, 1)\) | |
📉📈 The Graphs
Increasing, approaches 0 on the left
Decreasing, approaches 0 on the right
✏️ Examples of Exponential Functions
\(f(x) = 2^x\)
Base 2, increasing
\(f(x) = 3 \cdot 2^x\)
Vertical stretch by factor 3
\(f(x) = 2^x + 3\)
Shift 3 up, asymptote y=3
\(f(x) = e^x\)
Base e ≈ 2.718
🧮 Part B: Exponential Equations
Definition
An exponential equation is an equation in which the unknown appears in the exponent.
Examples:
\(2^x = 8\) , \(3^{2x+1} = 27\) , \(5^x = 5^{3x-4}\)
⭐ Core Solving Principle
\(a^{f(x)} = a^{g(x)} \implies f(x) = g(x)\)
(when \(a > 0, a \neq 1\))
💡 In words: if the bases are equal, then the exponents are equal!
🔧 Solving Methods
Method 1: Matching Bases
When it is possible to bring both sides to the same base.
Example: Solve \(2^x = 32\)
Solution:
Recognise that \(32 = 2^5\)
\(2^x = 2^5\)
Bases are equal, therefore: \(x = 5\)
Example: Solve \(9^x = 27\)
Solution:
Convert to a common base (3):
\(9 = 3^2\) and \(27 = 3^3\)
\((3^2)^x = 3^3\)
\(3^{2x} = 3^3\)
\(2x = 3\)
\(x = \frac{3}{2} = 1.5\)
Method 2: Using Logarithms
When it is not possible to bring both sides to the same base.
Example: Solve \(2^x = 5\)
Solution:
Apply log to both sides:
\(\log(2^x) = \log 5\)
\(x \cdot \log 2 = \log 5\)
\(x = \frac{\log 5}{\log 2} \approx \frac{0.699}{0.301} \approx 2.32\)
💡 General formula:
\(a^x = b \implies x = \frac{\log b}{\log a} = \log_a b\)
Method 3: Substitution (disguised quadratic)
When different powers of the same base appear.
Example: Solve \(4^x - 6 \cdot 2^x + 8 = 0\)
Solution:
Note that \(4^x = (2^2)^x = (2^x)^2\)
Substitute \(t = 2^x\) (where \(t > 0\)):
\(t^2 - 6t + 8 = 0\)
\((t-2)(t-4) = 0\)
\(t = 2\) or \(t = 4\)
Back to x:
\(2^x = 2 \implies x = 1\)
\(2^x = 4 \implies x = 2\)
Answer: \(x = 1\) or \(x = 2\)
⚠️ Important: always remember that \(t = a^x > 0\)!
If \(t \leq 0\) arises, that solution is invalid.
📐 Reminder: Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product of powers | \(a^m \cdot a^n = a^{m+n}\) | \(2^3 \cdot 2^4 = 2^7\) |
| Quotient of powers | \(\frac{a^m}{a^n} = a^{m-n}\) | \(\frac{2^5}{2^2} = 2^3\) |
| Power of a power | \((a^m)^n = a^{m \cdot n}\) | \((2^3)^2 = 2^6\) |
| Zero exponent | \(a^0 = 1\) | \(5^0 = 1\) |
| Negative exponent | \(a^{-n} = \frac{1}{a^n}\) | \(2^{-3} = \frac{1}{8}\) |
| Fractional exponent | \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\) | \(8^{\frac{2}{3}} = \sqrt[3]{64} = 4\) |
⚖️ Exponential Inequalities
Important rule: the direction of the inequality depends on the base!
\(a > 1\)
\(a^{f(x)} > a^{g(x)}\)
\(\Downarrow\)
\(f(x) > g(x)\)
Direction preserved
\(0 < a < 1\)
\(a^{f(x)} > a^{g(x)}\)
\(\Downarrow\)
\(f(x) < g(x)\)
Direction reverses!
✏️ Example: Solve \(2^x > 8\)
\(2^x > 2^3\)
Base 2 > 1, so the direction is preserved:
\(x > 3\)
✏️ Example: Solve \(\left(\frac{1}{2}\right)^x > 4\)
\(\left(\frac{1}{2}\right)^x > \left(\frac{1}{2}\right)^{-2}\) (because \(4 = 2^2 = \left(\frac{1}{2}\right)^{-2}\))
Base \(\frac{1}{2}\) < 1, so the direction reverses:
\(x < -2\)
💡 Tips for the Exam
1️⃣ Try a common base first
Always check whether you can bring both sides to the same base before using logs
2️⃣ Know your powers
Powers of 2: 2, 4, 8, 16, 32, 64...
Powers of 3: 3, 9, 27, 81...
3️⃣ Substitution t
When both \(a^{2x}\) and \(a^x\) appear – substitute \(t = a^x\)
4️⃣ Inequalities
Base > 1: direction preserved
Base < 1: direction reverses!
📝 Summary
Exponential function: \(f(x) = a^x\)
Solving principle: \(a^{f(x)} = a^{g(x)} \implies f(x) = g(x)\)
Using log: \(a^x = b \implies x = \log_a b\)
The exponential function and the logarithm are inverse functions!