exponential logarithmic functions
Overview
# Exponential and Logarithmic Functions This A-Level topic covers the fundamental properties of exponential functions (y = aˣ) and logarithmic functions (y = logₐx), including their relationship as inverse functions, graph transformations, and the laws of logarithms. Students learn to solve exponential and logarithmic equations, apply these functions to real-world contexts such as compound interest and population growth, and work with natural logarithms (ln x) and the exponential constant e. This topic is essential for Pure Mathematics papers, frequently appearing in both short-answer questions and extended problems that integrate differentiation, integration, and modelling applications.
Core Concepts & Theory
Exponential functions have the form f(x) = aˣ where a > 0, a ≠ 1. The most important exponential function is eˣ (Euler's number, e ≈ 2.71828), which appears naturally in calculus due to its unique derivative property: d/dx(eˣ) = eˣ.
Logarithmic functions are the inverse of exponential functions. For y = aˣ, we write x = log_a(y). The natural logarithm ln(x) is the inverse of eˣ, meaning ln(eˣ) = x and e^(ln x) = x for x > 0.
Key Properties of Logarithms:
- Product rule: log(AB) = log A + log B
- Quotient rule: log(A/B) = log A − log B
- Power rule: log(Aⁿ) = n log A
- Change of base: log_a(x) = ln(x)/ln(a)
- Special values: ln(1) = 0, ln(e) = 1
Important Equations:
- eˣ⁺ʸ = eˣ · eʸ
- (eˣ)ⁿ = eⁿˣ
- e⁰ = 1
Memory Aid (LOGS): L = Laws to Learn, O = One becomes zero (ln 1 = 0), G = Growth exponentially, S = Sum becomes product (ln(AB) = ln A + ln B)
The exponential graph y = eˣ passes through (0,1), increases rapidly, never touches the x-axis (horizontal asymptote y = 0). The logarithmic graph y = ln(x) is its reflection in y = x, passing through (1,0) with vertical asymptote x = 0, defined only for x > 0.
Detailed Explanation with Real-World Examples
Exponential growth models countless real phenomena. Population growth follows P(t) = P₀e^(kt) where P₀ is initial population, k is growth rate, and t is time. If bacteria double every hour, they exhibit exponential growth—not linear. After 10 hours, you don't have 10× the bacteria; you have 2¹⁰ = 1024× more!
Compound interest demonstrates exponential functions: A = P(1 + r/n)^(nt) where P is principal, r is annual rate, n is compounds per year, and t is years. Continuously compounded interest uses A = Pe^(rt), showing why 'e' matters in finance.
Radioactive decay follows N(t) = N₀e^(-λt) where λ is the decay constant. Half-life occurs when N(t) = N₀/2, giving t_(1/2) = ln(2)/λ. Carbon-14 dating uses this principle with half-life 5,730 years.
pH scale in chemistry is logarithmic: pH = -log₁₀[H⁺]. Each unit decrease means 10× more acidic. pH 3 is 100× more acidic than pH 5.
Sound intensity (decibels) uses L = 10log₁₀(I/I₀). A whisper (30 dB) to normal conversation (60 dB) represents a 1000× increase in intensity, not double!
Analogy: Think of exponential growth like compound interest on knowledge—each piece you learn multiplies your understanding, not just adds to it. Logarithms are like asking "how many times must I multiply to reach this number?" If 2³ = 8, then log₂(8) asks "what power gives 8?" Answer: 3.
Worked Examples & Step-by-Step Solutions
**Example 1:** Solve 3e^(2x-1) = 15 for x, giving your answer in exact form. *Solution:* 1. Divide both sides by 3: e^(2x-1) = 5 2. Take natural log: ln(e^(2x-1)) = ln(5) 3. Use inverse property: 2x - 1 = ln(5) 4. Solve for x: 2x = 1 + ln(5) 5. **x = (1 + ln 5)/2** ✓ *Examiner note:* Must show ln ...
Unlock 3 More Sections
Sign up free to access the complete notes, key concepts, and exam tips for this topic.
No credit card required · Free forever
Key Concepts
- Exponential Function: A function of the form f(x) = a^x, where a is a positive constant (a ≠ 1) and x is the exponent.
- Logarithmic Function: The inverse of an exponential function, written as log_a(x) = y if and only if a^y = x.
- Natural Logarithm (ln x): Logarithm to the base e (Euler's number), where e ≈ 2.71828.
- Common Logarithm (log x): Logarithm to the base 10.
- +3 more (sign up to view)
Exam Tips
- →Always check the domain of logarithmic functions: the argument of a logarithm must be strictly positive. This is crucial when solving equations and identifying extraneous solutions.
- →Be proficient with all laws of logarithms and indices. Many errors occur from misapplying these rules. Practice expanding and condensing expressions.
- +3 more tips (sign up)
More Mathematics Notes