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Easy Ways to Solve Log and Exponential Equations

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Max Luke

1/11/2023

Pre-Calculus

Solving Exponential & Logarithmic Equations

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Pre-Calculus

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Binomial Theorem

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The Binomial Theorem

Easy Ways to Solve Log and Exponential Equations

A comprehensive guide to solving logarithmic and exponential equations strategies with detailed examples and methods.

  • Master the fundamental approaches including isolation, condensing, and the common base method for solving exponential equations
  • Learn essential techniques for isolating and condensing logs in equations through step-by-step examples
  • Understand how to check for extraneous solutions and domain restrictions
  • Practice with various equation types including natural logarithms and exponential forms
  • Apply systematic problem-solving strategies across different logarithmic scenarios
...

1/11/2023

108

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 2: Advanced Logarithmic Equation Techniques

This section delves into more complex logarithmic equations and demonstrates various solution methods. The page emphasizes the importance of proper logarithm manipulation and condensing techniques.

Vocabulary: Condensing logarithms refers to combining multiple logarithmic terms into a single logarithm.

Example: In solving log₈x+2x+2 = log₈3x−23x-2, the equation can be solved by setting the arguments equal: x+2 = 3x-2.

Highlight: The solution process often involves converting logarithmic equations to exponential form or vice versa.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

View

Page 3: Natural Logarithms and Exponential Solutions

This page covers equations involving natural logarithms lnln and the exponential function ee. It demonstrates how to solve equations containing these special functions.

Definition: Natural logarithm lnln is the logarithm with base e, where e is Euler's number approximately2.71828approximately 2.71828.

Example: For ln3x+53x+5 = 8, the solution involves using e⁸ = 3x+5, then solving for x.

Highlight: Domain restrictions are particularly important when solving natural logarithmic equations.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

View

Page 4: Complex Exponential Equations

The focus here is on solving more complicated exponential equations, particularly those that result in quadratic equations after manipulation.

Example: For eˣ²+2x=0, the solution involves factoring ex−2eˣ-2ex−1eˣ-1=0 and solving two separate equations.

Vocabulary: Quadratic form in exponential equations occurs when the variable appears in two different powers.

Highlight: When solving exponential equations, converting to logarithmic form can often simplify the solution process.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

View

Page 5: Advanced Applications and Combined Techniques

This final page demonstrates complex problems that combine multiple concepts and techniques from previous sections.

Example: For logx2+1x²+1 = logx−2x-2 + logx+3x+3, the solution requires condensing logarithms and solving the resulting polynomial equation.

Highlight: Final verification of solutions is crucial to ensure they satisfy both the equation and domain restrictions.

Quote: "Check for extraneous solutions when solving logarithmic equations to ensure validity of answers."

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Subjects

Pre-Calculus

Binomial Theorem

The Binomial Theorem

 

Pre-Calculus

•

108

•

Jan 11, 2023

•

5 pages

Easy Ways to Solve Log and Exponential Equations

user profile picture

Max Luke

@maxluke

A comprehensive guide to solving logarithmic and exponential equations strategies with detailed examples and methods.

  • Master the fundamental approaches including isolation, condensing, and the common base method for solving exponential equations
  • Learn essential techniques for isolating and condensing logs in... Show more

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 2: Advanced Logarithmic Equation Techniques

This section delves into more complex logarithmic equations and demonstrates various solution methods. The page emphasizes the importance of proper logarithm manipulation and condensing techniques.

Vocabulary: Condensing logarithms refers to combining multiple logarithmic terms into a single logarithm.

Example: In solving log₈x+2x+2 = log₈3x−23x-2, the equation can be solved by setting the arguments equal: x+2 = 3x-2.

Highlight: The solution process often involves converting logarithmic equations to exponential form or vice versa.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 3: Natural Logarithms and Exponential Solutions

This page covers equations involving natural logarithms lnln and the exponential function ee. It demonstrates how to solve equations containing these special functions.

Definition: Natural logarithm lnln is the logarithm with base e, where e is Euler's number approximately2.71828approximately 2.71828.

Example: For ln3x+53x+5 = 8, the solution involves using e⁸ = 3x+5, then solving for x.

Highlight: Domain restrictions are particularly important when solving natural logarithmic equations.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 4: Complex Exponential Equations

The focus here is on solving more complicated exponential equations, particularly those that result in quadratic equations after manipulation.

Example: For eˣ²+2x=0, the solution involves factoring ex−2eˣ-2ex−1eˣ-1=0 and solving two separate equations.

Vocabulary: Quadratic form in exponential equations occurs when the variable appears in two different powers.

Highlight: When solving exponential equations, converting to logarithmic form can often simplify the solution process.

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 5: Advanced Applications and Combined Techniques

This final page demonstrates complex problems that combine multiple concepts and techniques from previous sections.

Example: For logx2+1x²+1 = logx−2x-2 + logx+3x+3, the solution requires condensing logarithms and solving the resulting polynomial equation.

Highlight: Final verification of solutions is crucial to ensure they satisfy both the equation and domain restrictions.

Quote: "Check for extraneous solutions when solving logarithmic equations to ensure validity of answers."

4.5 Notes Solving Thursday, September 23, 2021 Notes 9/23 4.5: Solving Log + Exp. Equations Strategies / Scenarios Isolate 4 (5) 9:47 AM log

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Page 1: Core Strategies for Solving Logarithmic and Exponential Equations

This page introduces the fundamental strategies for solving logarithmic and exponential equations. The content focuses on three main approaches: isolation of logarithms, comparing equal logarithms, and identifying common bases in exponential equations.

Definition: Logarithmic equations are equations that contain logarithms of variables, while exponential equations contain variables in the exponent.

Highlight: When solving logarithmic equations, always check for extraneous solutions due to domain restrictions.

Example: For exponential equations like bˣ = bʸ, the solution is x = y when using the common base method.

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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

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I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

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Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user