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Graphing Exponential and Logarithmic Functions

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Graphing Exponential and Logarithmic Functions

Study Guide

Key Definition

Exponential functions are of the form $y = b^x$ and have graphs that increase rapidly. Logarithmic functions are the inverses, of the form $y = \log_b{x}$.

Important Notes

The graph of $y = 2^x$ has the x-axis as an asymptote.
Replacing $x$ with $-x$ reflects the graph across the y-axis.
Always specify the base of a logarithm; do not assume a default base.
Exponential and logarithmic functions are inverses of each other.
Logarithmic functions have a vertical asymptote at $x = 0$.

Mathematical Notation

$y = b^x$ represents an exponential function

$y = \log_b{x}$ represents a logarithmic function

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

Remember to use proper notation when solving problems

Why It Works

Exponential functions grow by constant factors, and logarithmic functions increase their output by fixed amounts each time the input is multiplied by the same factor, reflecting their inverse relationship.

Remember

The graph of $y = b^x$ passes through (0,1) since any number to the power of zero is one.

Quick Reference

Exponential Function:$y = b^x$

Logarithmic Function:$y = \log_b{x}$

Understanding Graphing Exponential and Logarithmic Functions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Exponential functions grow by multiplying the previous value by the base, whereas logarithmic functions return the exponent needed to reach a given value from the base.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the base of the function $y = 3^x$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a bacteria culture that doubles every hour. If you start with 100 bacteria, how many will there be in 5 hours?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $y = \log_2{x}$ and $y = 3$, what is $x$?

Challenge Quiz

Single Choice Quiz

Advanced

For the function $y = \log_{10}{x}$, which of the following is true?

Recap

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Review key concepts and takeaways