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Logarithmic Function Manipulation

Alice White

Alice White

8 min read

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Study Guide Overview

This study guide covers logarithmic function manipulation, focusing on the relationship between logarithmic and exponential functions. Key properties include the Product, Power, and Change of Base rules, along with their graphical interpretations. It also emphasizes natural logarithms (ln) and their relationship to e. Finally, the guide provides practice questions and exam tips covering simplifying expressions, solving equations, graphing, and applications of logarithms.

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.
Memory Aid

Think of logs as "exponent finders." If by=xb^y = x, then log⁡b(x)=y\log_b(x) = y. This helps remember the relationship between exponential and log functions.

Properties of Logarithmic Functions 🧮

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

Key Concept

✖️ Product Property

  • Concept: The log of a product is the sum of the logs.
  • Formula: log⁡b(xy)=log⁡b(x)+log⁡b(y)\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.
Exam Tip

This property is super useful for expanding logarithmic expressions and can help simplify complex equations.

  • 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)f(x) = \log_b(kx) is equivalent to f(x)=log⁡b(k)+log⁡b(x)=a+log⁡b(x)f(x) = \log_b(k) + \log_b(x) = a + \log_b(x), where a=log⁡b(k)a = \log_b(k).

download (2).png

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 po...
Previous Topic - Logarithmic FunctionsNext Topic - Exponential and Logarithmic Equations and Inequalities

Question 1 of 11

If by=xb^y = x, which of the following is the equivalent logarithmic form? 🚀

y=log⁡x(b)y = \log_x(b)

x=log⁡b(y)x = \log_b(y)

y=log⁡b(x)y = \log_b(x)

b=log⁡y(x)b = \log_y(x)