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Solving Exponential Equations

Learn to solve exponential equations by applying exponent properties and logarithms.

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Solving Exponential Equations

Study Guide

Key Definition

An exponential equation has the form $a^x = b^y$, where $a>0$, $a≠1$, $b>0$, and $b≠1$. If $a = b$, then $x = y$.

Important Notes

Use the property of equality for exponential functions: $b^x = b^y \text{ implies } x = y$
If bases are different, convert them to the same base if possible
Use logarithms when bases cannot be rewritten to be the same
Check your solution by substituting back into the original equation
Remember that $b$ must be positive and not equal to 1

Mathematical Notation

$\times$ represents multiplication

$\div$ represents division

$x^2$ means x squared

$\sqrt{x}$ represents square root of x

$\log_b{x}$ is the logarithm of x to the base b

Remember to use proper notation when solving problems

Why It Works

By equating the exponents when bases are the same, you use the property of equality: $b^x = b^y \rightarrow x = y$.

Remember

Convert different bases to the same base when possible and use logarithms when necessary for solving.

Quick Reference

Property:$b^x = b^y \text{ if and only if } x = y$

Understanding Solving Exponential Equations

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

For equations of the form $a^x = a^y$ with $a>0$ and $a≠1$, the exponents must be equal. Example: to solve $3^x = 3^4$, set $x = 4$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Solve $4^{2x-1} = 64$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A population of bacteria doubles every hour. If you start with 100 bacteria, how many bacteria will there be after 3 hours?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $3^x = 27$, find $x$ without using a calculator.

Challenge Quiz

Single Choice Quiz

Advanced

Solve $2^{x+1} = 8^{x-1}$

Recap

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Review key concepts and takeaways