Solving Exponential Equations - Varsity Tutors

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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 bRemember 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

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

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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 $x = 4$.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

A$x = 2$B$x = 3$C$x = 4$D$x = 5$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A population of bacteria doubles every hour. If you start with 100 bacteria, how many bacteria will there be after 3 hours?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$x = 0$B$x = 1$C$x = 2$D$x = 3$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

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