#AP Pre-Calculus: Logarithmic Functions - Your Night-Before Guide 🚀

Hey there! Let's make sure you're feeling super confident about logarithmic functions for tomorrow's exam. We'll break it all down, keep it chill, and make sure everything clicks. Let's do this! 💪

#2.11 Logarithmic Functions: Unlocking the Inverses of Exponentials

Remember, logarithmic functions aren't some random math beast. They're actually the inverse of exponential functions. Think of them as the 'undo' button for exponentials. This connection is key! 💡

#Domain and Range: Where Log Functions Live

• Domain: Log functions, written as $y = \log_b(x)$, only accept positive numbers. So, x > 0. No zero or negative inputs allowed! 🙅
• Range: The output (y-values) can be any real number. Log functions are free to roam across the entire number line! ⛩️

Caption: Notice how the graph only exists for x > 0, but extends infinitely up and down.

#Extrema, Concavity, and Inflection Points: The Shape of Logs

• Increasing/Decreasing: If the exponential function is increasing, its inverse log function is also increasing (and vice versa). Log graphs are either always going up or always going down. 📈

Caption: Base > 1 means the graph goes up; base < 1 means it goes down.

• Concavity: Log functions are always either concave up or concave down. No switching! 🫡
• Extrema & Inflection Points: Because of this consistent behavior, log functions don't have maximums, minimums, or inflection points (unless you're looking at a restricted interval). 🪂

Caption: Notice the different concavity and direction combinations.

#Additive Transformations: Shifting the Graph

• Horizontal Shifts: The function $g(x) = f(x + k)$ shifts the graph of $f(x)$ horizontally by k units. So, $g(x) = \log_b(x + k)$ moves the log graph left or right. ➕
• Logarithmic Check:
  • If input values of $g(x) = f(x + k)$ aren't proportional over equal output intervals, then g is not logarithmic. ❌
  • If output values of $g(x) = f(x + k)$ are proportional over equal input intervals, then f is logarithmic. ✅

Caption: The graph of log base 3 of x is shifted left by 4 units.

#Limits and Asymptotes: Approaching Infinity

• Vertical Asymptote: Log functions have a vertical asymptote at x = 0 (for the basic form $y = \log_b(x)$). This means the graph gets super close to the y-axis but never touches it. 🛑
• End Behavior: As x approaches 0 from the right, y approaches either positive or negative infinity. As x approaches infinity, y also approaches either positive or negative infinity. ⛓️

Caption: The arrows indicate the behavior as x approaches 0 and infinity.

• Limits:
  • $\lim_{x \to 0^+} a\log_b(x) = \pm \infty$
  • $\lim_{x \to \infty} a\log_b(x) = \pm \infty$

#Final Exam Focus 🎯

Okay, let's nail down what's super important for the exam. Focus on these areas:

• Domain and Range: Know them cold! This is a frequent MCQ topic.
• Transformations: Be ready to shift log graphs left, right, up, or down.
• Asymptotes and End Behavior: Understand how the function behaves as x approaches 0 and infinity.
• Connecting to Exponential Functions: Remember that logs are inverse of exponentials. This connection is often tested.

#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: Watch out for domain restrictions! Log functions don't like negative numbers or zero.
• FRQ Strategy: Show all your work, even if you think it's obvious. Partial credit is your friend!

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. What is the domain of the function $f(x) = \log_2(x - 3)$? (A) $x > 0$ (B) $x > 3$ (C) $x < 3$ (D) All real numbers

2. Which of the following describes the end behavior of $g(x) = -\log_3(x)$ as x approaches infinity? (A) Approaches positive infinity (B) Approaches negative infinity (C) Approaches zero (D) Oscillates

3. The graph of $y = \log_b(x)$ passes through the point (9, 2). What is the value of b? (A) 3 (B) 81 (C) $\sqrt{3}$ (D) $\frac{1}{3}$

#Free Response Question

Consider the function $h(x) = 2\log_4(x + 1) - 3$.

(a) State the domain of h(x). (b) Describe the transformations applied to $y = \log_4(x)$ to obtain the graph of h(x). (c) Find the equation of the vertical asymptote of h(x). (d) Sketch the graph of h(x), clearly showing the asymptote and at least two points.

Scoring Breakdown:

(a) 1 point: Correct domain (b) 2 points: 1 point for each correct transformation (vertical stretch and shift) (c) 1 point: Correct vertical asymptote equation (d) 3 points: 1 point for asymptote, 1 point for correct shape, 1 point for at least two points

#Answers

Multiple Choice:

1. (B) $x > 3$
2. (B) Approaches negative infinity
3. (A) 3

Free Response:

(a) Domain: $x > -1$ (b) Transformations: Vertical stretch by a factor of 2, horizontal shift left by 1 unit, vertical shift down by 3 units. (c) Vertical Asymptote: $x = -1$ (d) Sketch should show a vertical asymptote at $x=-1$, and a curve that passes through at least two points, such as (0, -3) and (3, -1)

You've got this! Go ace that exam! 🌟