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Solving Logarithmic Equations

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Solving Logarithmic Equations

Study Guide

Key Definition

A logarithm answers the question: 'To what power must we raise the base to obtain a given number?' For example, in $\log_b a = x$, it means $b^x = a$.

Important Notes

Logarithms are the inverse of exponents.
If there is no base number written, it is assumed to be 10.
You can convert to exponential form or use logarithm properties (product, quotient, power rules) to simplify equations before solving.
Always check your solutions in the original equation.

Mathematical Notation

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

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Logarithms are the inverse of exponents, allowing us to solve equations involving exponential growth or decay by converting them to a more manageable form.

Remember

One useful approach is to express logarithmic equations in exponential form or use properties of logarithms to simplify before solving.

Quick Reference

Logarithm to Exponential:$\log_b a = x \Rightarrow b^x = a$

Understanding Solving Logarithmic Equations

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When you see $\log_{10} x = y$, rewrite as $10^y = x$. For example, $\log_{10} 100 = y$ becomes $10^y = 100$, so $y = 2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $\log_2 8 = x$, what is $x$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Solve for $x$ in the equation $\log_5 (x+1) = 2$.

Challenge Quiz

Single Choice Quiz

Advanced

Which equation is equivalent to $\log_3 27 = x$?

Recap

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Review key concepts and takeaways