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Dec 1, 2025

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Understanding Exponential and Logarithmic Equations

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Jezriel Ortiz

@jezrielortiz

These logarithmic and exponential equations might look intimidating at first,... Show more

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1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

Solving Complex Logarithmic Equations

Ever wonder how to crack those tough log equations with multiple variables? This problem shows you the power of smart substitution and factoring.

The key breakthrough here is recognizing that 4x2+4x+1=(2x+1)=n\sqrt{4x^2+4x+1} = (2x+1) = n and 6x2+11x+4=(3x+4)(2x+1)=m2⋅n6x^2+11x+4 = (3x+4)(2x+1) = m^2 \cdot n. Once you substitute these expressions, the original messy equation becomes much cleaner: log⁡mn+log⁡n(m2⋅n)=4\log_m n + \log_n (m^2 \cdot n) = 4.

Setting log⁡mn=a\log_m n = a transforms this into a simple equation: a+2=4a + 2 = 4, so a=2a = 2. But don't forget to check if a=1a = 1 works too! When you solve both cases, you get potential solutions x=−3x = -3, x=34x = \frac{3}{4}, and x=−1x = -1.

Quick Tip: Always check your answers in the original equation - some solutions might not work due to domain restrictions!

1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

Function Notation and Series in Logarithms

Here's where logarithms meet series - and it's actually pretty cool! The function L(n)=log⁡2(an)L(n) = \log_2(\sqrt[n]{a}) might look complex, but it simplifies beautifully.

Using the power rule for logarithms, L(n)=1nlog⁡2aL(n) = \frac{1}{n}\log_2 a. This means 1L(n)=nlog⁡2a\frac{1}{L(n)} = \frac{n}{\log_2 a}. The sum 1L(1)+1L(2)+...+1L(10)=77\frac{1}{L(1)} + \frac{1}{L(2)} + ... + \frac{1}{L(10)} = 77 becomes 1+2+...+10log⁡2a=77\frac{1 + 2 + ... + 10}{\log_2 a} = 77.

The magic number here is that 12+22+...+102=3851^2 + 2^2 + ... + 10^2 = 385, which gives us 385log⁡2a=77\frac{385}{\log_2 a} = 77. Solving this equation step by step leads to a=21/5a = 2^{1/5}, but wait - there's an error in the original work. The correct answer is a=21/5a = 2^{1/5}, not 32.

The exponential equation in problem 3 uses substitution brilliantly. Let y=22xy = 2^{2x}, solve the quadratic y2−65y+64=0y^2 - 65y + 64 = 0, and get x=0x = 0 or x=3x = 3.

Pro Strategy: When you see repeated exponential expressions, try substitution to create a simpler equation!

1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

Inequalities and System Solving

Exponential inequalities can trip you up if you forget one crucial rule: when the base is less than 1, flip the inequality sign!

In the inequality 2x+2−2x+3−2x+4>5x+1−5x+22^{x+2} - 2^{x+3} - 2^{x+4} > 5^{x+1} - 5^{x+2}, factoring out common terms gives us 2x(−20)>5x(−20)2^x(-20) > 5^x(-20). After dividing by −20-20 (and flipping the inequality), we get (25)x<1\left(\frac{2}{5}\right)^x < 1. Since 25<1\frac{2}{5} < 1, this means x>0x > 0.

The system of equations in problem 5 looks brutal, but substitution saves the day again. Let x=2ax = 2^a and y=log⁡2by = \log_2 b, then solve the resulting linear system. You'll find x=4x = 4 and y=3y = 3, which means a=2a = 2 and b=8b = 8.

Therefore, a2+b2=4+64=68a^2 + b^2 = 4 + 64 = 68. Clean and straightforward once you set it up right!

Memory Hook: Always remember that dividing by a negative number flips inequality signs - this rule applies to exponential inequalities too!



We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

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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

iOS user

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

Android user

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

iOS user

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

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

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

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

iOS user

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

Android user

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

iOS user

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

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

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

 

Math (SAT®)

•

93

•

Dec 1, 2025

•

3 pages

Understanding Exponential and Logarithmic Equations

user profile picture

Jezriel Ortiz

@jezrielortiz

These logarithmic and exponential equations might look intimidating at first, but they're actually just puzzles waiting to be solved! Once you master a few key techniques like substitution and factoring, you'll be tackling these problems with confidence.

1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

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Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Solving Complex Logarithmic Equations

Ever wonder how to crack those tough log equations with multiple variables? This problem shows you the power of smart substitution and factoring.

The key breakthrough here is recognizing that 4x2+4x+1=(2x+1)=n\sqrt{4x^2+4x+1} = (2x+1) = n and 6x2+11x+4=(3x+4)(2x+1)=m2⋅n6x^2+11x+4 = (3x+4)(2x+1) = m^2 \cdot n. Once you substitute these expressions, the original messy equation becomes much cleaner: log⁡mn+log⁡n(m2⋅n)=4\log_m n + \log_n (m^2 \cdot n) = 4.

Setting log⁡mn=a\log_m n = a transforms this into a simple equation: a+2=4a + 2 = 4, so a=2a = 2. But don't forget to check if a=1a = 1 works too! When you solve both cases, you get potential solutions x=−3x = -3, x=34x = \frac{3}{4}, and x=−1x = -1.

Quick Tip: Always check your answers in the original equation - some solutions might not work due to domain restrictions!

1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Function Notation and Series in Logarithms

Here's where logarithms meet series - and it's actually pretty cool! The function L(n)=log⁡2(an)L(n) = \log_2(\sqrt[n]{a}) might look complex, but it simplifies beautifully.

Using the power rule for logarithms, L(n)=1nlog⁡2aL(n) = \frac{1}{n}\log_2 a. This means 1L(n)=nlog⁡2a\frac{1}{L(n)} = \frac{n}{\log_2 a}. The sum 1L(1)+1L(2)+...+1L(10)=77\frac{1}{L(1)} + \frac{1}{L(2)} + ... + \frac{1}{L(10)} = 77 becomes 1+2+...+10log⁡2a=77\frac{1 + 2 + ... + 10}{\log_2 a} = 77.

The magic number here is that 12+22+...+102=3851^2 + 2^2 + ... + 10^2 = 385, which gives us 385log⁡2a=77\frac{385}{\log_2 a} = 77. Solving this equation step by step leads to a=21/5a = 2^{1/5}, but wait - there's an error in the original work. The correct answer is a=21/5a = 2^{1/5}, not 32.

The exponential equation in problem 3 uses substitution brilliantly. Let y=22xy = 2^{2x}, solve the quadratic y2−65y+64=0y^2 - 65y + 64 = 0, and get x=0x = 0 or x=3x = 3.

Pro Strategy: When you see repeated exponential expressions, try substitution to create a simpler equation!

1 log √4x²+4x+1 + log₂ (6x²+11x+4)= 4; m= √3x+4¸n=2x+1 ; find x 1. Soln |4x²+4x+1 = (2x+1) (2x+1) =√ non =√n² = n 2 6x² +11x+4 = (3x+4) (2x+

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Inequalities and System Solving

Exponential inequalities can trip you up if you forget one crucial rule: when the base is less than 1, flip the inequality sign!

In the inequality 2x+2−2x+3−2x+4>5x+1−5x+22^{x+2} - 2^{x+3} - 2^{x+4} > 5^{x+1} - 5^{x+2}, factoring out common terms gives us 2x(−20)>5x(−20)2^x(-20) > 5^x(-20). After dividing by −20-20 (and flipping the inequality), we get (25)x<1\left(\frac{2}{5}\right)^x < 1. Since 25<1\frac{2}{5} < 1, this means x>0x > 0.

The system of equations in problem 5 looks brutal, but substitution saves the day again. Let x=2ax = 2^a and y=log⁡2by = \log_2 b, then solve the resulting linear system. You'll find x=4x = 4 and y=3y = 3, which means a=2a = 2 and b=8b = 8.

Therefore, a2+b2=4+64=68a^2 + b^2 = 4 + 64 = 68. Clean and straightforward once you set it up right!

Memory Hook: Always remember that dividing by a negative number flips inequality signs - this rule applies to exponential inequalities too!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

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App Store

4.8/5

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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

iOS user

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

Android user

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

iOS user

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

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

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

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

iOS user

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

Android user

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

iOS user

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

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

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