Unit 2: Exponential and Logarithmic Functions

Overview

• Focus on exponential and logarithmic functions
• Important for AP exam
• Includes AP-style questions at the end of each topic

Sequences

• Arithmetic Sequences:

  • Linear functions with a common rate of change
  • Common difference, e.g., 2
  • Equations:
    • (a_n = a_0 + dn)
    • (a_n = a_k + d(n-k))

• Geometric Sequences:

  • Increase proportionally
  • Common ratio, e.g., 3
  • Equations:
    • (g_n = g_0 \cdot r^n)
    • (g_n = g_k \cdot r^{n-k})

Exponential Functions

• Properties:
  • Equation: (f(x) = ab^x)
  • (a \neq 0), (b > 0), (b \neq 1)
  • Growth: (a > 0, b > 1)
  • Decay: (a > 0, 0 < b < 1)
  • Always increasing or decreasing (concave)
  • No points of inflection

Rules of Exponents

• Product Property: (b^m \cdot b^n = b^{m+n})
• Power Property: ((b^m)^n = b^{mn})
• Negative Exponent Property: (b^{-n} = \frac{1}{b^n})
• Exponent Root Property: (b^{1/k} = \sqrt[k]{b})

Modeling Exponential Functions

• Use scenarios involving multiplication
• Requires two points to derive function
• Systems of equations for modeling
• (e) (approx. 2.718) for continuous growth/decay
• Calculator regressions for exponential functions

Modeling with Functions

• Linear: (y = mx + b)
• Quadratic: (y = a(x - b)^2 + c)
• Exponential: (y = ab^x)
• Residual plots: Assess model fit, aim for random distribution

Inverse Functions

• Swap x and y to find inverses
• Must be one-to-one (pass horizontal line test)

Logarithms

• Logarithmic Expressions:
  • (b^a = c) becomes (\log_b{c} = a)
  • Common log: base 10
  • Used for solving exponential equations

Logarithmic Functions

• Inverse of exponential functions
• Swap domain and range with exponential
• Graph is a reflection over (y = x)

Properties of Logarithms

• Product Property: (\log_b(xy) = \log_b{x} + \log_b{y})
• Quotient Property: (\log_b\left(\frac{x}{y}\right) = \log_b{x} - \log_b{y})
• Power Property: (\log_b(x^k) = k \cdot \log_b{x})
• Change of Base Formula: Use common logs for base conversion
• Natural log: (\ln{x} = \log_e{x})

Modeling Logarithmic Functions

• Use when X's increase proportionally (multiplicative growth)
• Logarithmic regression on calculator to find best fit

Semi-Log Plots

• Used to identify if data follows exponential or logarithmic trends
• Logarithmic scaling on one axis makes exponential/logarithmic functions appear linear

Conclusion

• The unit covers essential topics for calculus and AP exams
• Importance of understanding both exponential and logarithmic functions and their properties
• Encourages practice with AP-style questions and use of graphing tools

Note: This unit is challenging, but mastering it will aid in understanding more complex mathematical concepts in future topics.