Exponential & Logarithmic Functions
Simplify complex log expressions, master exponential equations, and learn why checking the domain is the single most important step when solving logarithmic equations.
1. The Core Concept
The Power of Logs: Logarithms are the mathematical inverses of exponents. If
aʸ = x, then y = logₐ x. In the DSE, you must be able to use the fundamental laws of logarithms to condense expressions and solve equations where the variable is trapped in an exponent (e.g., aˣ = b).2. DSE Exponential & Logarithmic Functions Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Laws of Logarithms | • Product: logₐ(xy) = logₐ x + logₐ y• Quotient: logₐ(x/y) = logₐ x − logₐ y• Power: logₐ(xⁿ) = n logₐ x• Identity: logₐ a = 1 and logₐ 1 = 0 | Before applying any log laws, always verify that the bases are identical (e.g., both are base 2 or base 10). |
| 2. Change of Base | logₐ b = log_c b / log_c a = log b / log a | Use this to convert any base to base 10 (common log) so you can compute the values on your DSE-approved calculator. |
| 3. Exponential Equations | • Same Base: aᶠ⁽ˣ⁾ = aᵍ⁽ˣ⁾ ⟹ f(x) = g(x)• Different Base: Take log of both sides: aˣ = b ⟹ x log a = log b ⟹ x = log b / log a | Quadratic Form: If you see p(a²ˣ) + q(aˣ) + r = 0, substitute u = aˣ to solve the quadratic equation pu² + qu + r = 0. |
| 4. Logarithmic Equations | 1. Condense both sides: logₐ f(x) = logₐ g(x) ⟹ f(x) = g(x)2. Convert to exponent: logₐ f(x) = y ⟹ f(x) = aʸ | You must check your answers: Log arguments cannot be ≤ 0. Plug solutions back into the original equations to reject extraneous roots. |
| 5. Inverse Graphs | • Exponential: y = aˣ (passes through (0, 1), asymptote y = 0)• Logarithmic: y = logₐ x (passes through (1, 0), asymptote x = 0) | Both graphs are reflections of each other along the diagonal line y = x. |
Click the graph to explore it interactively →
💡 Exam Traps & Secrets
⚠️ The "Fake Log Laws" Trap (Extremely Common!)Examiners deliberately design multiple-choice questions to trick students who invent non-existent laws. Avoid these common traps:
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log(a + b) ≠ log a + log b (This expression cannot be simplified further)❌
log a / log b ≠ log(a − b) (The quotient of two logs is a change of base, log_b a, not subtraction)❌
(log a)ⁿ ≠ n log a (The power must be on the argument aⁿ, not on the entire log expression)⚠️ The Domain / Extraneous Root TrapWhen solving a logarithmic equation like
log₂(x + 3) + log₂(x − 1) = 5, you might find a negative value of x (e.g., x = −5) after solving. Rule: Always plug your solved x values back into the original log arguments (the expressions inside the brackets). Since the domain of any log must be strictly positive (x + 3 > 0 and x − 1 > 0), any solution that makes an argument ≤ 0 must be rejected. Failing to reject extraneous roots will cost you the final accuracy mark in Paper 1.⚠️ The Negative Quadratic Substitution Trap (u = aˣ)When solving
a²ˣ − 5aˣ − 6 = 0, substituting u = aˣ gives u = 6 or u = −1. Since an exponential curve y = aˣ is always strictly positive (aˣ > 0), the negative solution aˣ = −1 has no real solution and must be rejected. Only solve for aˣ = 6 ⟹ x = log 6 / log a.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Condensing log expressions using product and quotient laws)Simplify
log₂ 48 − log₂ 3 + log₂(1/4).(Solving different-base exponential equations by taking logs)Solve
3²ˣ⁺¹ = 7ˣ, giving your answer correct to 3 significant figures.(Combining log terms, solving a quadratic equation, and verifying the domain to reject extraneous roots)Solve
log₂(x + 3) + log₂(x − 1) = 5.