Logarithmic Functions

Alice White

7 min read

Next Topic - Logarithmic Function Manipulation

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Study Guide Overview

This study guide covers logarithmic functions, focusing on their relationship as the inverse of exponential functions. Key topics include: domain and range, extrema, concavity, and inflection points, additive transformations (horizontal shifts), limits and asymptotes, and connections to exponential functions. The guide also provides practice questions and exam tips.

#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! ⛩️

Function y=log_2(x) graphed on a coordinate plane.

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. 📈

The graph on the left displays the function f(x)=log_b(x) with b > 1 and the curve is upward. The graph on the right displays the function f(x)=log_b(x) and 0 < b < 1 and the curve is downward.

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

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