IB Mathematics AA SL Exponential & logarithmic functions Study Notes

IB Mathematics AA SL Exponential & logarithmic functions Study Notes

IB Mathematics AA SL Exponential & logarithmic functions Study Notes Offer a clear explanation of Exponential & logarithmic functions , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Exponential & logarithmic functions

Exponential Functions and Their Graphs

Exponential Functions and Their Graphs

An exponential function has the form \( f(x) = b^x \) where \( b > 0 \) and \( b \neq 1 \).

  • \( f(x) = b^x \) grows rapidly if \( b > 1 \) (exponential growth).
  • \( f(x) = b^x \) decays if \( 0 < b < 1 \) (exponential decay).
  • \( f(x) = e^x \) is a natural exponential function, where \( e \approx 2.718 \).

Key Features:

  • Domain: \( (-\infty, \infty) \)
  • Range: \( (0, \infty) \)
  • y-intercept at \( (0,1) \)
  • Horizontal asymptote at \( y = 0 \)

Financial Applications:

Exponential functions model compound interest and geometric sequences and series:

\( A = P \left(1 + \frac{r}{n}\right)^{nt} \) or \( A = P e^{rt} \)

where: A = final amount, P = principal, r = rate, n = number of compounding periods per year, t = time

Example: Sketch the graphs of \( f(x) = 2^x \) and \( f(x) = e^x \).

▶️Answer/Explanation
  • Both graphs pass through \( (0, 1) \).
  • Both have horizontal asymptote \( y = 0 \).
  • Both increase as \( x \to \infty \).
  • Both approach zero as \( x \to -\infty \).

Example: A sum of ₹1000 is invested at 5% annual interest, compounded continuously. Find the amount after 10 years.

▶️Answer/Explanation
  • Formula: \( A = P e^{rt} \)
  • \( P = 1000 \), \( r = 0.05 \), \( t = 10 \)
  • \( A = 1000 \cdot e^{0.05 \times 10} \)
  • \( A \approx 1000 \cdot e^{0.5} \approx 1000 \cdot 1.6487 = 1648.72 \)

The amount after 10 years is approximately ₹1648.72.

Logarithmic Functions and Their Graphs

Logarithmic Functions and Their Graphs

A logarithmic function is the inverse of an exponential function. It is of the form:

  • \( f(x) = \log_b x \), where \( b > 0 \) and \( b \neq 1 \)
  • \( f(x) = \ln x \), where \( \ln x = \log_e x \) and \( e \approx 2.718 \)

Key Features:

  • Domain: \( (0, \infty) \)
  • Range: \( (-\infty, \infty) \)
  • x-intercept at \( (1, 0) \)
  • Vertical asymptote at \( x = 0 \)
  • Passes through \( (b, 1) \) for base \(b \)

Graph Shape:

  • If \( b > 1 \): graph increases slowly as \( x \) increases
  • If \( 0 < b < 1 \): graph decreases as \( x \) increases

Example: Sketch the graph of \( f(x) = \ln x \).

▶️Answer/Explanation

Domain: \( x > 0 \)

Range: all real numbers

Vertical asymptote: \( x = 0 \)

Intercept: \( (1, 0) \)

Passes through: \( (e, 1) \)

Slow growth as \( x \to \infty \)

Relationships & Links to Other Subjects

Relationships Between Exponential and Logarithmic Functions

There are important relationships linking exponential and logarithmic functions, especially when converting between different bases.

  • Exponential identity:

$ a^x = e^{x \ln a} $ This expresses \( a^x \) in terms of base \( e \).

  • Logarithmic identity:

$ \log_a (a^x) = x \quad \text{(where } a > 0, a \neq 1 \text{ and } x \in \mathbb{R}) $ The logarithm of a power of \( a \) gives the exponent.

  • Exponential and logarithmic inverse relationship:

$ a^{\log_a x} = x \quad \text{(where } x > 0 \text{ and } a > 0, a \neq 1 \text{)} $ Exponentiating a logarithm returns the original number.

These relationships are useful for solving exponential and logarithmic equations and changing bases.

Links to Other Subjects

Exponential and logarithmic functions have important applications in various subjects across science:

    • Physics: Radioactive decay is modeled by an exponential decrease: $ N(t) = N_0 e^{- \lambda t} $ where \( N_0 \) is the initial quantity and \( \lambda \) is the decay constant.
    • Physics: Charging and discharging capacitors in RC circuits follow exponential growth or decay: $ V(t) = V_0 (1 – e^{-t/RC}) \quad \text{(charging)} $ $ V(t) = V_0 e^{-t/RC} \quad \text{(discharging)} $
    • Chemistry: First-order reaction rates use exponentials: $ [A](t) = [A]_0 e^{-kt} $ where \( k \) is the rate constant.
    • Chemistry: Activation energy calculations often use logarithms via the Arrhenius equation: $ k = A e^{-E_a/(RT)} $ or its log form for linearization: $ \ln k = \ln A – \frac{E_a}{R} \frac{1}{T} $
    • Biology: Growth curves (e.g., populations, bacteria) often follow exponential models: $ P(t) = P_0 e^{rt} $ where \( r \) is the growth rate.

These models allow us to analyze real-world processes mathematically using exponential and logarithmic functions.

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