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Logarithmic Functions

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Logarithmic Functions

Study Guide

Key Definition

A logarithmic function is an inverse of an exponential function, typically expressed as $y = \log_b x$ where $b$ is the base and $x$ is the argument.

Important Notes

The domain of $\log_b x$ is $x > 0$
The range of $\log_b x$ is all real numbers
Common logarithm uses base 10: $\log x$
Natural logarithm uses base e: $\ln x$
Logarithms convert multiplication into addition

Mathematical Notation

$\log_b x$ represents a logarithm with base $b$

Remember to use proper notation when solving problems

Why It Works

Logarithms help solve exponential equations by converting them to linear form, making calculations easier. For example, $b^y = x$ becomes $y = \log_b x$.

Remember

Key formulas include $\log_b (mn) = \log_b m + \log_b n$ and $\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n$

Quick Reference

Change of Base Formula:$\log_b x = \frac{\log_k x}{\log_k b}$

Understanding Logarithmic Functions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Logarithms are the inverse of exponentials, simplifying complex calculations into simple addition or subtraction.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\log_{10} 100$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're measuring sound intensity. If the sound intensity I is 1000 times the reference intensity I₀, what is the decibel level? The decibel level is calculated using $\log_{10}$ of the ratio of the intensity to a reference level.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider how $\log_b (xy) = \log_b x + \log_b y$ helps in simplifying multiplication.

Challenge Quiz

Single Choice Quiz

Advanced

Solve for the positive value of $x$ in the equation $\log_2 (x^2 - 1) = 3$.

Recap

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Review key concepts and takeaways