#AP Pre-Calculus: Logarithmic Function Properties - The Night Before 🌠

Hey there! Let's get you feeling confident about logarithmic functions. Remember, they're just the inverse of exponential functions, so if you're comfy with those, this will be a breeze! Let's dive in!

#2.12 Logarithmic Function Manipulation

#Quick Recap ⏰

• Logarithmic functions are inverses of exponential functions. They "undo" exponential operations.
• Just like exponential functions, logarithmic functions have properties that allow us to simplify, solve, and model equations.

#Properties of Logarithmic Functions 🧮

Let's explore the key properties that'll help you ace those exam questions!

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• Concept: The log of a product is the sum of the logs.
• Formula: $\log_b(xy) = \log_b(x) + \log_b(y)$
• In Simple Terms: When you multiply inside a log, you can split it into the sum of two logs.

• Graphical Interpretation: A horizontal dilation (stretch/compression) of a log function is equivalent to a vertical translation (shift up/down).
  • $f(x) = \log_b(kx)$ is equivalent to $f(x) = \log_b(k) + \log_b(x) = a + \log_b(x)$, where $a = \log_b(k)$.

The product property in action: Notice how multiplying the input (x) by 2 results in a vertical shift of the graph.

#🦸🏽 Power Property

• Concept: The log of a number raised to a power is the power times the log of the number.
• Formula: $\log_b(x^n) = n\log_b(x)$
• In Simple Terms: When you have an exponent inside a log, you can bring it down as a multiplier.

• Graphical Interpretation: Raising the input of a log function to a power results in a vertical dilation of the graph.
  • $f(x) = \log_b(x^k)$ results in a vertical stretch or compression.

The power property: Notice how the exponent of the input (x) becomes a multiplier of the log function.

Visualizing the power property: The graph of $f(x) = 2\log_4(x)$ is a vertical stretch of $y = \log_4(x)$.

#🔄 Change of Base Property

• Concept: Allows you to convert a log from one base to another.
• Formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$
• In Simple Terms: Change the base of a log by dividing by the log of the old base with the new base.

The change of base formula: This formula allows you to calculate logarithms with any base using a calculator that only has base 10 or base e logarithms.

• Graphical Interpretation: All log functions are vertical dilations of each other. Changing the base affects the "height" of the graph.

#Natural Logs and e 🌿

• Definition: The natural logarithm, denoted as $\ln(x)$, is the logarithm with base e, where e ≈ 2.71828. - Relationship: $\ln(x) = \log_e(x)$
• Inverse: $\ln(x)$ is the inverse of $e^x$.

• Domain: All positive real numbers.
• Range: All real numbers.

Comparing common and natural logarithms: Notice the difference in the base and how it affects the graph.

#Final Exam Focus 🎯

#High-Priority Topics

• Product, Power, and Change of Base Properties: Know these inside and out! They're fundamental for solving equations and simplifying expressions.
• Natural Logarithms: Understand their properties and how they relate to exponential functions.
• Graphical Transformations: Be able to relate the algebraic properties to changes in the graph of a log function.

#Common Question Types

• Simplifying Logarithmic Expressions: Use the properties to combine or expand logs.
• Solving Logarithmic Equations: Apply properties to isolate the variable.
• Graphing Logarithmic Functions: Understand how transformations affect the graph.
• Applications: Solve real-world problems using logarithmic models.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. Move on and come back if you have time.
• Common Pitfalls: Be careful with the order of operations and applying properties correctly.
• Strategies: If you're stuck, try rewriting the problem using a different property or a different base.

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Practice Question

Practice Questions

#Multiple Choice Questions

1. Simplify the expression: $\log_2(16x^3)$ (A) $4 + 3\log_2(x)$ (B) $4\log_2(x)$ (C) $16 + 3\log_2(x)$ (D) $4 + \log_2(x^3)$

2. Solve for x: $\log_3(x^2) = 4$ (A) 2 (B) 9 (C) 81 (D) 3

3. Which expression is equivalent to $\log_5(25) + \log_5(125)$? (A) $\log_5(150)$ (B) $\log_5(3125)$ (C) $5$ (D) $7$

#Free Response Question

Consider the function $f(x) = 2\ln(x+3) - 1$.

(a) State the domain of $f(x)$. (b) Find the x-intercept of $f(x)$. (c) Describe the transformations applied to the graph of $y = \ln(x)$ to obtain the graph of $f(x)$. (d) Sketch the graph of $f(x)$, labeling at least one key point.

Scoring Breakdown:

(a) [1 point] - 1 point for stating the correct domain: $x > -3$ or $(-3, \infty)$

(b) [2 points] - 1 point for setting $f(x) = 0$ and attempting to solve for x. - 1 point for the correct x-intercept: $x = e^{1/2} - 3$ or approximately -1.35

(c) [3 points] - 1 point for vertical stretch by a factor of 2. - 1 point for horizontal shift to the left by 3 units. - 1 point for vertical shift down by 1 unit.

(d) [2 points] - 1 point for a graph with the correct general shape of a logarithmic function. - 1 point for labeling a key point (e.g., x-intercept or a point from the domain).

#Answers

Multiple Choice:

1. (A) $4 + 3\log_2(x)$
2. (B) 9
3. (C) 5

Free Response:

(a) Domain: $x > -3$ or $(-3, \infty)$

(b) Set $f(x) = 0$: $2\ln(x+3) - 1 = 0$. Solve for x: $x = e^{1/2} - 3$

(c) The graph of $y = \ln(x)$ is vertically stretched by a factor of 2, horizontally shifted to the left by 3 units, and vertically shifted down by 1 unit.

(d) Sketch of the graph should show a log function with the correct transformations (e.g., a point around (-2, -1) and x-intercept at $e^{1/2} - 3$ )

You've got this! Remember to breathe and trust your preparation. You're ready to rock this exam! 🎉