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Logarithms

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Logarithms

Study Guide

Key Definition

A logarithm answers the question: "To what power do I have to raise $b$ to get $x$?" Expressed as $\log_b x = a$ means $b^a = x$.

Important Notes

Logarithms are the inverse of exponents.
Common bases are 10 and e.
The expression $\log_b x$ is read as "log base $b$ of $x$."
Use properties of exponents to simplify logarithmic expressions.
Logarithmic and exponential functions are inverse functions.

Mathematical Notation

$\log_b x = a$ means $b^a = x$

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Logarithms provide a way to solve for the exponent in exponential equations, allowing us to work backwards from a result to the power needed.

Remember

The base of the logarithm is the same as the base of the exponent in the equivalent exponential form.

Quick Reference

Logarithm Property:$\log_b (xy) = \log_b x + \log_b y$

Power Property:$\log_b (x^n) = n \cdot \log_b x$

Understanding Logarithms

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Logarithms are the inverse of exponents. If $b^a = x$, then $\log_b x = a$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\log_2 8$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If the population of a town is $1000$ and it triples every decade, how long will it take to reach a population of $27000$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\log_5 x = 2$, what is $x$?

Challenge Quiz

Single Choice Quiz

Advanced

Solve for $x$: $\log_2 (x^2) = 6$

Recap

Watch & Learn

Review key concepts and takeaways