#AP Precalculus: Unit 2 Study Guide - Exponential and Logarithmic Functions

Welcome to your ultimate guide for Unit 2! This unit is all about exponential and logarithmic functions, which are super important for modeling real-world stuff. Let's get this bread! 🍞

This unit is a big deal, so make sure you understand these concepts well. You'll see them pop up in both multiple-choice and free-response questions.

#🧭 Unit 2 Overview: Exponential and Logarithmic Functions

This unit focuses on understanding and applying exponential and logarithmic functions. These functions are essential for modeling growth, decay, and various other phenomena. You'll learn to graph, analyze, and manipulate these functions, and see how they connect to sequences, composite functions, and inverse functions.

• Exponential functions: Form $f(x) = ab^x$, where 'a' and 'b' are constants, and b > 0 and not equal to 1. They model situations with growth or decay proportional to the current value.

• Logarithmic functions: In the form $f(x) = \log_b(x)$, they are the inverse of exponential functions. They model situations where the rate of change is inversely proportional to the current value.

You'll also learn to:

• Graph and analyze these functions, including their asymptotes, domain, and range.
• Solve exponential and logarithmic equations.
• Use properties of logarithms to simplify expressions.
• Use semi-log plots to analyze exponential data.

#🔨 Unit Breakdown

#🔷 Arithmetic & Geometric Sequences

Sequences are foundational to understanding patterns and growth. Let's break them down:

• Arithmetic Sequences: Each term differs by a constant amount (common difference, 'd').

  • Formula: $u_n = a + (n-1)d$, where 'a' is the first term.
  • Example: 1, 3, 5, 7... (d = 2)

• Geometric Sequences: Each term is multiplied by a constant amount (common ratio, 'r').

  • Formula: $u_n = ar^{n-1}$, where 'a' is the first term.

  • Example: 2, 6, 18, 54... (r = 3)

#↪️ Exponential & Logarithmic Functions

These are the stars of the show! 🌟

• Exponential Functions: $f(x) = ab^x$

  • 'a' is the initial value, and 'b' is the growth/decay factor.
  • Graph is a curve that increases rapidly (if b > 1) or decreases rapidly (if 0 < b < 1).
  • Used to model population growth, radioactive decay, compound interest, etc.

• Logarithmic Functions: $f(x) = \log_b(x)$

  • The inverse of exponential functions.
  • Graph approaches the y-axis asymptotically.
  • Used to measure sound intensity, pH levels, earthquake magnitudes, etc.

  

#⭕️ Composite Functions

Combining functions is like a recipe! 🧑‍🍳

• Composite Functions: Combining two or more functions where the output of one is the input of the other.

  • Notation: $(f \circ g)(x) = f(g(x))$, where 'g' is the inner function and 'f' is the outer function.
  • Order matters! $f(g(x))$ is generally not the same as $g(f(x))$.

  

#🔄 Inverse Functions

Functions that 'undo' each other! 🔄

• Inverse Functions: If $f(g(x)) = x$ and $g(f(x)) = x$, then $f(x)$ and $g(x)$ are inverses.

  • Notation: The inverse of $f(x)$ is $f^{-1}(x)$.
  • To find the inverse, swap x and y in the original function and solve for y.

  

#🪵 Semi-log Plots

A special type of graph for exponential data! 📈

• Semi-log Plots: Use a logarithmic scale on one axis and a linear scale on the other.

  • Useful for visualizing data with a wide range of values.
  • Exponential data appears as a straight line on a semi-log plot.
  • The slope of the line on the semi-log plot can be used to fit an exponential model to the data.

  

#🎯 Final Exam Focus

Alright, let's focus on what's most likely to appear on the exam:

• High-Priority Topics:

  • Exponential and logarithmic function graphs and properties.
  • Solving exponential and logarithmic equations.
  • Understanding and applying inverse functions.
  • Interpreting semi-log plots.
  • Arithmetic and geometric sequences

• Common Question Types:

  • Graphing exponential and logarithmic functions and identifying key features (asymptotes, intercepts, domain, range).
  • Solving equations involving exponential and logarithmic functions.
  • Applying exponential and logarithmic functions to model real-world scenarios (e.g., growth, decay, compound interest).
  • Finding composite and inverse functions.
  • Analyzing data using semi-log plots.
  • Questions that combine multiple concepts from different units.

• Last-Minute Tips:

  • Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
  • Common Pitfalls: Pay close attention to the order of operations, especially with composite functions. Remember the difference between inverse functions and reciprocal functions.
  • Strategies: Read the questions carefully and identify what they are asking. Draw diagrams or graphs to visualize the problem. Double-check your work before submitting.

#📝 Practice Questions

Practice Question

#Multiple Choice Questions

1. What is the inverse of the function $f(x) = 2x + 3$? (A) $f^{-1}(x) = \frac{x-3}{2}$ (B) $f^{-1}(x) = \frac{x+3}{2}$ (C) $f^{-1}(x) = \frac{1}{2x+3}$ (D) $f^{-1}(x) = 2x-3$

2. The population of a town grows exponentially at a rate of 5% per year. If the initial population is 1000, what will be the population after 10 years? (A) 1500 (B) 1628 (C) 1629 (D) 1640

3. Which of the following is the graph of a logarithmic function with a base greater than 1? (A) A graph that increases rapidly from the origin (B) A graph that approaches the x-axis asymptotically as x increases (C) A graph that decreases rapidly from the origin (D) A graph that approaches the y-axis asymptotically as x increases

#Free Response Question

A bacterial culture grows exponentially. Initially, there are 200 bacteria. After 3 hours, there are 800 bacteria.

a) Write an equation that models the number of bacteria, B, after t hours.

b) How many bacteria will there be after 6 hours?

c) How long will it take for the number of bacteria to reach 3200?

Scoring Breakdown:

a) (3 points)

• 1 point for recognizing the exponential growth model $B(t) = ab^t$
• 1 point for finding the initial value a = 200
• 1 point for finding the growth factor b = 2 (or equivalent)

b) (1 point)

• 1 point for correctly calculating the number of bacteria after 6 hours using the equation from part a)

c) (2 points)

• 1 point for setting up the equation to solve for t
• 1 point for correctly solving for t

You've got this! Go ace that exam! 💪