#AP Pre-Calculus: Inverses of Exponential Functions - Your Night-Before-the-Exam Guide

Hey there! Let's make sure you're feeling awesome about exponential and logarithmic functions for your AP Pre-Calculus exam. This guide is designed to be super clear and helpful, especially when you're doing that last-minute review. Let's get started!

#2.10 Inverses of Exponential Functions

#What's the Big Deal? 🤔

We're diving into logarithmic functions and how they're totally connected to exponential functions. Think of them as two sides of the same coin—or better yet, as inverses of each other! Understanding this relationship is key for the exam. 🔑

#Defining Logarithmic Functions

A logarithmic function looks like this: $f(x) = a \log_b(x)$.

• b is the base of the logarithm. It has to be greater than 0 and not equal to 1. * a is the coefficient of the function. It cannot be 0. ### Exponential Functions: A Quick Review

An exponential function is written as: $f(x) = ab^x$.

• a is the coefficient.
• b is the base.

#The Inverse Relationship

• Exponential Growth: Output values change multiplicatively as input values change additively.
  
• Logarithmic Growth: Output values change additively as input values change multiplicatively.

Think of it like this: Exponential functions grow super fast, while logarithmic functions grow much slower. They undo each other's work!

#Reflections and the Identity Function

#The Identity Function: h(x) = x

This is just a straight line with a slope of 1 that passes through the origin. It's like a mirror for our functions. 🪞

#Visualizing the Inverse

• The graph of $f(x) = \log_b(x)$ is a reflection of the graph of $g(x) = b^x$ over the line h(x) = x.
  
• This is because they are inverse functions. The x and y values are swapped.

#Key Characteristics

#Exponential Function: $g(x) = b^x$

• Curve increases rapidly as x increases (if b > 1).
  
• Vertical asymptote at x = 0.
  
• No horizontal asymptote.

#Logarithmic Function: $f(x) = \log_b(x)$

• Curve increases slowly as x increases (if b > 1).
  
• Horizontal asymptote at y = 0.
  
• No vertical asymptote.

#Ordered Pairs: The Swap

If (s, t) is an ordered pair of the exponential function $g(x) = b^x$, then (t, s) is an ordered pair of the logarithmic function $f(x) = \log_b(x)$.

• Exponential: $t = b^s$ (input s, output t)
  
• Logarithmic: $s = \log_b(t)$ (input t, output s)

Practice Question

#Practice Questions

#Multiple Choice Questions

1. The graph of $y = 2^x$ is reflected over the line $y=x$. Which of the following is the equation of the resulting graph? (A) $y = -2^x$ (B) $y = \frac{1}{2}^x$ (C) $y = \log_2(x)$ (D) $y = \log_{\frac{1}{2}}(x)$

2. If the point (3, 8) lies on the graph of $y = b^x$, which of the following points must lie on the graph of $y = \log_b(x)$? (A) (8, 3) (B) (3, -8) (C) (-3, 8) (D) (-8, -3)

#Free Response Question

Consider the functions $f(x) = 3^x$ and $g(x) = \log_3(x)$.

(a) Sketch the graphs of $f(x)$ and $g(x)$ on the same coordinate plane. Include the line $y=x$.

(b) Identify the domain and range of both $f(x)$ and $g(x)$.

(c) If the point (2, 9) lies on the graph of $f(x)$, what point must lie on the graph of $g(x)$? Justify your answer.

Scoring Rubric:

(a) Graph (3 points)

• 1 point: Correct graph of $f(x) = 3^x$
• 1 point: Correct graph of $g(x) = \log_3(x)$
• 1 point: Correctly drawn line $y=x$

(b) Domain and Range (2 points)

• 1 point: Correct domain and range for $f(x)$ (Domain: $(-\infty, \infty)$, Range: $(0, \infty)$)
• 1 point: Correct domain and range for $g(x)$ (Domain: $(0, \infty)$, Range: $(-\infty, \infty)$)

(c) Point and Justification (2 points)

• 1 point: Correct point (9, 2)
• 1 point: Justification that the coordinates are swapped due to the inverse relationship

#Answers

Multiple Choice:

1. (C)
2. (A)

Free Response: (a) Graph should show $f(x) = 3^x$ as an increasing exponential curve, $g(x) = \log_3(x)$ as an increasing logarithmic curve, and the line $y=x$. The logarithmic curve should be a reflection of the exponential curve across the line $y=x$.
(b) For $f(x) = 3^x$: Domain: $(-\infty, \infty)$, Range: $(0, \infty)$. For $g(x) = \log_3(x)$: Domain: $(0, \infty)$, Range: $(-\infty, \infty)$.
(c) The point (9, 2) must lie on the graph of $g(x)$ because exponential and logarithmic functions are inverses of each other, so their ordered pairs are reversed.