Crash Course — Unit 2: Exponential and Logarithmic Functions

• Arithmetic vs. Geometric Sequences
  €” Arithmetic sequences change by adding a constant (like adding $5 each time), while geometric sequences change by multiplying by a constant (like doubling each time).
• Linear vs. Exponential Functions
  €” These are the continuous versions of the sequences above. Linear functions have a constant rate of change (a straight line), while exponential functions have a constant percent or proportional rate of change (a curve).
• The Form f(x) = ab^x — This is your template for exponential functions. a is the initial value (the y-intercept), and b is the growth/decay factor.
• Function Composition f(g(x)) — This means plugging the output of one function (g(x)) into the input of another function (f). The order matters!
• Inverse Functions
  €” An inverse function, f⁻¹(x), swaps the inputs and outputs of the original function to "undo" it. Graphically, it's a reflection over the line y=x.
• Logarithms are Exponents
  €” The expression log_b(y) simply asks the question: "What exponent do I put on base b to get the value y?"
• Logarithm Properties
  €” These are your tools for rewriting and simplifying log expressions. They let you turn multiplication into addition, division into subtraction, and exponents into coefficients.
• Solving Exponential & Log Equations
  €” You use the inverse operation to solve. To solve an exponential equation, take the log of both sides. To solve a log equation, exponentiate both sides.
• Model Validation from Data
  €” To decide if a data set is linear or exponential, check the outputs. If the differences are constant, it's linear. If the ratios are constant, it's exponential.
• Semi-log Plots
  €” A special type of graph where one axis has a logarithmic scale. If you plot exponential data on a semi-log plot, it will form a straight line, which is a great way to confirm your model.

Key Formulas / Terms

• Exponential Function
  f(x) = a * b^x

  • a = initial value (when x=0)
  • b = growth/decay factor. If b > 1, it's growth. If 0 < b < 1, it's decay.
• Growth/Decay Factor from Rate
  b = 1 + r (for growth) or b = 1 - r (for decay), where r is the rate as a decimal.
• Logarithm Definition (The Conversion Circle)
  log_b(y) = x is the exact same statement as b^x = y.
• Logarithm Properties
  • Product Rule: log_b(M * N) = log_b(M) + log_b(N)
  • Quotient Rule: log_b(M / N) = log_b(M) - log_b(N)
  • Power Rule: log_b(M^p) = p * log_b(M)
• Change of Base Formula
  log_b(a) = log(a) / log(b) (Use this to find any log on your calculator).

Exam Traps

• Trap
  Confusing growth factor (b) with growth rate (r). A bank account growing by 3% has a rate r = 0.03, but its growth factor is b = 1 + 0.03 = 1.03. The formula uses b, not r! · Counter: Always ask yourself: "Am I given the multiplier, or the percent change?" Convert the percent change (r) to the multiplier (b) before you build your function.
• Trap
  Forgetting to check for extraneous solutions in log equations. The argument of a logarithm (the stuff in the parentheses) must be positive. · Counter: After solving a log equation, always plug your solution back into the original equation. If it makes any log's argument zero or negative, you have to discard it.
• Trap
  Misapplying log properties, especially with addition. Students often write log(x + 5) = log(x) + log(5). This is completely wrong. · Counter: Burn this into your brain: Log properties only work for multiplication, division, and powers inside a single logarithm. There is no property for the log of a sum or difference.
• Trap
  Confusing f(g(x)) with g(f(x)) or f(x) * g(x). Composition is not multiplication, and the order is critical. · Counter: Work from the inside out. For f(g(2)), first find the value of g(2). Then, take that result and plug it into f.
• Trap
  Incorrectly identifying the horizontal asymptote. In y = a * b^x + k, the asymptote is y=k, not always y=0. · Counter: The horizontal asymptote is always determined by the vertical shift (k). If there's no number added or subtracted at the end, then k=0 and the asymptote is y=0.