Tuesday, November 29, 2016

High School Math Solutions – Inequalities Calculator, Logarithmic Inequalities

Last post, we talked about radical inequalities. In this post, we will talk about how to solve logarithmic inequalities. We’ll see logarithmic inequalities in forms such as \log_b(f(x))<a or \ln(⁡f(x))<a. In order to solve these inequalities, the goal will be to isolate the variable, just as in any inequality, and we will do this by getting rid of the log function. Let’s dive in and see how to solve logarithmic inequalities.

Steps to solve logarithmic inequalities:
1. Use algebraic manipulation to move anything that is not in the logarithmic expression to one side
2. Combine logarithmic expressions
3. Isolate the variable by getting rid of the logarithmic expression
             Ex:    \log_b⁡(f(x))<a

b^(\log_b⁡(f(x))) <b^a

f(x)<b^a

Ex: \ln(f(x))<a

\ln⁡(f(x))<\ln⁡(e^a )

f(x)<e^a
4. Solve inequality
5. Get the range for the expression in the original log function
             Ex:  \log_b⁡(f(x))
Range: f(x)>0
6. Combine ranges

Let’s do an example step by step now.

First example (click here):

                                            \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

Step 1: Use algebraic manipulation to move anything that is not in the logarithmic expression to one side

There’s nothing to move, so we can skip this step.

Step 2: Combine logarithmic expressions

                                            \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

                                                      \log_4⁡(\frac{x+3}{x+2})≥\frac{3}{2}


Step 3: Isolate the variable by getting rid of the logarithmic expression

                                                 \log_4⁡(\frac{x+3}{x+2})≥\frac{3}{2}

                                                4^(\log_4⁡(\frac{x+3}{x+2})) ≥4^(\frac{3}{2})

                                                         \frac{x+3}{x+2}≥8

Step 4: Solve inequality

                                                          \frac{x+3}{x+2}≥8

We can see that this is now a rational inequality. We won’t solve this step by step; I will show the answer after solving this inequality. If you are struggling with solving this inequality, visit the blog post on rational inequalities.

                                                             -2<x≤\frac{-13}{7}

Step 5: Get the range for the expression in the original log function

                                                   \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

                                                       x+3>0              x+2>0

                                                       x>-3              x>-2

Step 6: Combine ranges

                                                    -2<x≤\frac{-13}{7},    x>-2,    x>-3

                                                              -2<x≤\frac{-13}{7}

That wasn’t too bad! Let’s see some more examples.


Second example (click here):



Last example (click here):




Solving logarithmic inequalities is not too difficult. Just remember to get the ranges inside the logarithmic expressions and to double check your work. For more help and practice on this topic visit Symbolab’s  practice.

Until next time,

Leah


102 comments:

  1. Lol, no doubt math is really a tough subject for many people and in high school, it becomes tougher, here is an advice keep practicing and you will do better.

    ReplyDelete
  2. This took me to my school days. Sites like this are really good. Thanks

    UAE Company Registration

    ReplyDelete
  3. You need to examine this site for some info or ideas on how to write good compare and contrast essay. This could be really useful for your future

    ReplyDelete
  4. This is a wonderful article. I would like to suggest you that please keep on sharing such type of information with us. I really found it to much informative. It is what i was searching from many days piknu

    ReplyDelete
  5. Take me back to the mountainside. Under the northern lights, chasing after the stars When we are full of life and the night is calm, there is no fear like five nights at freddy's fnaf

    ReplyDelete
  6. Logarithms and exponential are inverse operations. In other words, one operation undoes the other. For example, 102 = 100 and log 100 = 2. Also, 10 log x = x. Note: log is base 10. Professional Essay Writers Online - essaytigers

    ReplyDelete
  7. This Is Really Great Work. Thank You For Sharing Such A Good And Useful Information Here In The Blog slipcovers neos overshoes canada

    ReplyDelete
  8. We are usually available 24/7 and our fast essay writing service team can be contacted and also seek our pay for term papers at any time.

    ReplyDelete
  9. Very informative and helpful at the same time, appreciated the knowledge you shared with others, the content is lit,looking for some more informative content to come, keep up the good work, keep spreading knowledge, thank you!
    Alex Roe The 5th Wave

    ReplyDelete
  10. thank you.
    คาสิโนออนไลน์ที่น่าเชื่อถือและมีความเป็นมืออาชีพที่สุดในตอนนี้
    โปรโมชั่นGclub ของทางทีมงานตอนนี้แจกฟรีโบนัส 50%
    เพียงแค่คุณสมัคร สล็อตออนไลน์ กับทางทีมงานของเราเพียงเท่านั้น
    ร่วมมาเป็นส่วนหนึ่งกับเว็บไซต์คาสิโนออนไลน์ของเราได้เลยค่ะ
    สมัครสล็อตออนไลน์ >>> Goldenslot
    สนใจร่วมสนุกกับ คาสิโนออนไลน์ คลิ๊กได้เลย
    มีทั้งคาสิโนออนไลน์ หวยออนไลน์ ฟุตบอลออนไลน์ สล็อตออนไลน์ และอื่นๆอีกมากมาย

    ReplyDelete
  11. I am a website designer and developer from Australia. I have deep interests in Website design, development, marketing, math and astrophysics in-spite of the fact that I graduated as a software developer.

    ReplyDelete
  12. Great Blog with useful information. student Faces many problems while doing Assignment. Solve student Academic Problem.
    cheap assignment help
    cheap assignment help Australia

    ReplyDelete
  13. This Is Really Great Work. Thank You For Sharing Such A Good And Useful Information Here In The Blog

    Trademark registration service dubai

    ReplyDelete
  14. Hey, I’m David. I’m a web developer living in New York. I am a fan of technology, web development, and fashion. I’m also interested in

    writing and photography. You can visit my company website with a click on the button above.
    geeksquad.com
    geek squad tech support

    ReplyDelete
  15. fast social follower will help you buy the best social media services such as Buy Facebook Followers . At the best prices and all the real services , real follower , real likes etc .

    ReplyDelete
  16. Roku provides the simplest way to stream entertainment to your TV. On your terms. With thousands of available channels to choose from https://ww-roku.com.

    ReplyDelete
  17. Activate Roku.com/link, go to https://ww-roku.com/link account enter Roku com link code displayed on Roku TV. My roku com link not working use new Roku code

    ReplyDelete
  18. https://r-roku.com this is simplest way to stream entertainment to your TV On your terms. With thousands of available channels to choose from for more details.

    ReplyDelete
  19. Roku is a streaming device that allows you to watch movies, videos, Web series, etc. on your TV via the internet. You can create your Roku account at https://r-roku.com/link. Roku has 3000+ channel content with online apps with your roku account.

    ReplyDelete
  20. Roku Streaming Device, you can watch the biggest content of channels on your TV, mobile, laptop, etc. You can select your favorite channels with your Roku account via https://www-roku.us.

    ReplyDelete
  21. McAfee Activate! you can get started activation with enter product code/key at https://mamcafee.com/activate and visit for more details.

    ReplyDelete
  22. Webroot antivirus software quickly block viruses, malware, spyware or cyber threats other malicious files attempting damage your data. visit http://enter-webroot.com to get this amazing antivirus.

    ReplyDelete
  23. Microsoft office is a group of office applications. Each application have a unique purpose and each applications offers a specific services to the users. Visit here https://officecom-setup.uk.com to download office setup.

    ReplyDelete
  24. Norton antivirus is the most preferred and highly recommended antivirus. Norton Antivirus provides overall protection to the device without damaging . http://n-nortoncom.com/setup remove infected sites, online browsing viruses, Trojans, and other harmful software from the device.

    ReplyDelete
  25. Roku provides the simplest way to stream entertainment to your TV. On your terms. With thousands of available channels to choose from.
    Roku.com/link

    ReplyDelete
  26. Roku provides the simplest Roku.com/link way to stream entertainment to your TV. On your terms. With thousands of available channels to choose from.

    ReplyDelete
  27. Visit Roku link Roku.com/link to enter Roku activation code and activate your roku device on Tv.Contact for Support if facing any problem

    ReplyDelete
  28. Roku link - visit Roku.com/link link to activate Roku device and setup your Roku player by using roku activation code. Contact roku support for any queries.

    ReplyDelete
  29. Sign in to your Roku account Roku.com/link . A Roku account gives you access to an amazing selection of movies, TV shows, music and more from the Roku Channel Store.

    ReplyDelete
  30. mcafee activate - Mcafee.com/activate Shielding the computing devices against dangerous online threats, malware, phishing scams, Trojan horses and other infections.

    ReplyDelete
  31. Download and Activate your mcafee product via Mcafee.com/activate activate. If facing any issue read the blog and help yourself with the procedure.

    ReplyDelete
  32. Norton Setup Norton.com/setup allows you to deploy your Norton setup effectively. Learn the basics to download, install and activate Norton Setup without any error.

    ReplyDelete
  33. Create an account or sign in at Norton.com/setup setup, redeem the product and download Norton setup. After installation, enter the product key for activation

    ReplyDelete
  34. setup & Activate roku link Roku.com/link Streaming device. Get exhaustive guidelines to Enter Roku Code in Roku account from the experts at roku link.

    ReplyDelete
  35. office setup - Here you can download latest version of Office Office.com/setup the link says 2016 but the version is most updated just visit office setup.

    ReplyDelete
  36. McAfee activate - Find the process for downloading Mcafee.com/activate , installing, and activation McAfee products on PC, Mac, Android, and iOS. To activate the product, visit mcafee activate.

    ReplyDelete
  37. Office setup with your product key. Get started: step 1. Go to Office.com/setup setup - Sign in, 2nd. Enter product key, and 3rd. Install Office 365 or Office 2019.

    ReplyDelete
  38. This comment has been removed by the author.

    ReplyDelete
  39. This comment has been removed by the author.

    ReplyDelete
  40. Norton Security gives you the best of Norton for the numerous ways you interface. www.norton.com/setup With a solitary basic arrangement, you get complete insurance that is made to verify your preferred gadgets—PCs, Macs®, cell phones and tablets—so you can be sheltered in any case and any place you associate.

    ReplyDelete
  41. Norton Security Real-time insurance against existing and norton setup developing malware including ransomware and infections.

    ReplyDelete
  42. Microsoft Office is an application software. www.office.com/setup It is used in most of the businesses and organizations Install office setup on your Pc's to get best advantage of it. mostly used tasks are ms-word, Ms-excel and Ms-powerpoint.

    ReplyDelete
  43. It is safe to say that you are on the Hulu (No Ads) or Hulu (No Ads) + Live TV plan? hulu.com/activate Assuming this is the case, presently you can download a great many TV shows and motion pictures on your bolstered iOS gadget to watch disconnected later.

    ReplyDelete
  44. Would you like to set up hulu antivirus in your PC?... at that point click here activate hulu for more subtleties.

    ReplyDelete
  45. Roku spilling players need web access to stream content. www.roku.com/link They utilize remote to interface with your home system.

    ReplyDelete
  46. Enact Roku connect, go to roku link code account enter Roku interface code showed on Roku TV. My roku com interface not working utilize new Roku code.To download click for more details..

    ReplyDelete

  47. You need a Roku record to enact your gadget and access diversion. www.roku.com/link There are no month to month gear rental expenses. Including an installment technique lets you effectively lease/purchase motion pictures or buy in to well known administrations.

    ReplyDelete
  48. Norton Security Premium gives all what Norton Security does, in addition to it backs up the individual records, photographs, www.norton.com/setup recordings and different recollections on your PC.

    ReplyDelete
  49. Norton Security exist With proactive web insurance, cell phone following, norton setup propelled protection innovation and the sky is the limit from there, it's the one answer for assistance keep you, and your gadgets, safe.

    ReplyDelete
  50. Mind blowing, lightweight, fused protection for PC, Mac and Android, webroot.com/safe cloud-based Webroot Internet Security Complete with antivirus guarantees singular information by thwarting the latest malware, phishing, and advanced ambushes. Visit to present webroot antivirus in your pc.

    ReplyDelete
  51. Download and introduce Webroot Software and www.webroot.com/safe AntiVirus which influence your PC and workstation.

    ReplyDelete
  52. Ms Office suite offers a wide selection of products but to access a licensed version, you must have www.office.com/setup. After you exceed the free trial period, it becomes mandatory to purchase an Office setup product key. Get office setup here

    ReplyDelete
  53. Norton Security ensures your gadgets and the data on them, norton.com/setup VPN encodes the information you send and get when utilizing open Wi-Fi.

    ReplyDelete
  54. Roku gadgets are easy to set-up and simple to-utilize. roku link code They accompany a straightforward remote, and incredible highlights like Roku Search which makes it easy to discover what you need to watch.

    ReplyDelete
  55. Thank you for a detailed Article. Got to learn a lot of important things . For more information visit my website :- Yoga Meditation Retreat

    ReplyDelete
  56. You’re a very skilled blogger. I have joined your feed and look forward to seeking more of your fantastic post. Also, I have shared your web site in my social networks!
    Mcafee.com/activate
    Norton.com/setup
    Mcafee.com/activate
    Office.com/setup
    Mcafee.com/activate

    ReplyDelete
  57. office.com/setup
    - Nowadays businesses are not restricted by geographical barriers, which calls for a universally used program in every office premise. This is where Microsoft Office comes in as it is used widely in the offices all around the globe. Every industry and sector avails the esteemed apps and features of MS Office; be it schools, hospitals, offices, or banks. Microsoft Office also known as Office is a family of premium customer software manufactured and distributed by Microsoft.

    ReplyDelete
  58. You’re no longer beholden to cable subscription and if you want a wide range of television options get Roku device today start watching TV on upon Roku device activation by visiting roku.com/link.

    roku.com/link

    ReplyDelete

  59. Roku is a streaming device, which provides the simplest way to stream entertainment channel to your TV. On your terms. You can access more than 500000+ movies and TV episodes across free and pay on Roku. Also, you will get the best quality video streaming with it. You can watch shows, movies, an entertainment channel, web series, etc. For the activation of the Roku device, Go to URL: roku.com/link.

    roku.com/link

    ReplyDelete

  60. Roku is a package of entertainment. The Roku channel is the Expansion of programming for new generation children. Roku declares plans to Combined kids and family movies and TV together with the channel's other content, including subscriptions, its free, television, ad-supported movies, and live TV.

    roku.com/link

    ReplyDelete

  61. After the launching of "Kids & Family" section on Roku channel, Roku is also adding new features to give more control to parents over the channel, what their kids can watch when accessing Roku channels.

    roku.com/link

    ReplyDelete
  62. Sign in to enter your item key, get to your record, deal with your membership, and stretch out your Norton security to PC, Mac, Android, and iOS gadgets. On the off chance that you don't as of now have a Norton account, make one today. Welcome to Norton . Sign in to enter your item key, get to your record, deal with your membership, and stretch out your Norton security to PC, Mac, Android, and iOS gadgets.For more information visit norton.com/setup

    ReplyDelete
  63. norton.com/setup
    -Norton is undoubtedly the best antivirus software developed by the Symantec Corporation. It is available for every device like Mac, Windows, and even Android, as well. Most importantly, it offers lots of advanced features that include three types of scans (Quick, Full system, and Custom scan), and speak to the real person in support as well.

    ReplyDelete
  64. mcafee.com/activate-McAfee is an America based multinational enterprise that provides top-notch computer security programs. The need for computer security software has increased with the surge of internet users worldwide. McAfee offers premium-class web-security programs that help your system stay protected from various online threats.

    ReplyDelete

  65. Hulu provides the simplest way to stream entertainment to your TV. On your terms. With thousands of available channels. We provide information regarding Hulu Activation Code. Go to hulu.com/activate, and enter the Activation Code. To activate a new Hulu account using an existing email account and you can stream television shows and movies from the Hulu site using the Hulu activation link.

    www.hulu.com/activate

    ReplyDelete
  66. office setup, Vonage can help you find the right system to fit your needs & budget. We'll get you started quickly on setting up an office. Visit www.office.com/setup for more updates.

    ReplyDelete
  67. Norton Security has been giving the best on-line security answers for ensuring the clients' contraptions other than as information against the web threats. to get more about norton click norton setup now.

    ReplyDelete
  68. mcafee.com/activate - Shielding the computing devices against dangerous online threats, malware, phishing scams, Trojan horses and other infections.

    ReplyDelete
  69. Microsoft Office install has required the removal of the previously installed version of your Office product on the device or system. office setup and other subscription offers the various features, which you do not get when you do not purchase the Office product. The office can be used free, as Microsoft provides the trial versions of every tool.

    ReplyDelete
  70. Mcafee offers you the mcafee activate option to give best Home network protection, Multi-device protection, Web Protection, and Identity Theft.

    ReplyDelete
  71. Fast & Easy PC Virus Protection. Download avg.com/retail Now. Stops Viruses, Spyware, Ransomware and Other Malware. Easy Installation. Install in moments. Light & easy installation. Family protection. Protects forever. Stops viruses. Unsafe web link warnings.

    ReplyDelete
  72. Antivirus is the need of PCs that makes them infection free and we are going to give you full help to get the best antivirus introduced in your PCs and PCs. Download norton antivirus bundle with us - www.norton.com/setup for more details visit today.

    ReplyDelete
  73. Real-time problems and outages for Office 365. Is the service down? Can't login or connect? Here you see what is going on. for more details visit office.com/setup today.

    ReplyDelete
  74. To activate Hulu Plus on your device, either use the on-screen keyboard to enter your Hulu log in information or go to www.hulu.com/activate and enter the device activation code. Either of these methods will allow you to use Hulu on any Hulu-supported device.

    ReplyDelete
  75. We provide help & support for Roku setup & account login, when Roku not connecting & Remote not working. visit www.roku.com/link today.

    ReplyDelete
  76. Activate roku.com/link on your device, Chromecast, Apple TV, Playstation, or Xbox device and start watching free movies and TV shows.

    ReplyDelete
  77. office.com/setup will help you to deploy the office setup successfully on your device. Here are easy steps to download, install and activate office setup. Visit office setup now.

    ReplyDelete
  78. It is extremely important to protect your PCs, Mac, computers as well as mobile devices and tablets with www.avg.com/retail .

    ReplyDelete
  79. mcafee antivirus is broadly utilized antivirus identifies and kill PC infection, the mail worms,the trojan programs,and additionally enables your framework to free of infection and other malware is a significant every day challenge. for more subtleties visit: mcafee.com/activate today.

    ReplyDelete
  80. To get started with your Office Installation you must need valid product key code & visit: office.com/setup for futher issues.

    ReplyDelete
  81. Download and introduce norton setup Sign In to Norton. On the off chance that you don't have a Norton account, click Create an Account and complete the sign up procedure. In the Get Started window, click Enter a New Product Key. To take a crack at Automatic Renewal Service for your Norton membership.

    ReplyDelete
  82. Antivirus is the need of computers that makes them virus free and we are going to give you full support to get the best antivirus installed in your computers and laptops. Download norton antivirus package with us - norton.com/setup today.

    ReplyDelete
  83. Webroot is very well known for its immense security from the spyware, viruses. In order to protect your system from the spyware or any other viruses,visit: webroot.com/safe today.

    ReplyDelete
  84. office.com/setup - Most companies require an ultimate program to manage and create patterned data for their business performance and records. Even if you are a working individual or entrepreneur, you can still use Office for managing your work report and daily performance, such as using Excel to maintain your daily or monthly business revenue or profit. Moreover, you can use Excel spreadsheets to record your daily business expenses, and then you can refine your budget and deals to get the best ROI. Office is a productivity tool that can help you in understanding, analyzing, recording essential business data. 

    ReplyDelete
  85. Webroot.com/safe is the only security solution which can protect your system from viruses, worms, malware threats. Get Webroot safe today & install on your system and say goodbye to viruses.webroot.com/safe

    ReplyDelete
  86. If you want beautiful flowers then we have you covered. Flowers are for the soul, and it helps you communicate and convey your feeling as well. Therefore whether you are trying to wish someone on their special day or trying to surprise your wife, we at Captain Shop Sydney would be the perfect same day fresh flower delivery option as well.

    ReplyDelete
  87. AVG is the one of best antivirus which protects your computer and network from bad infection and threats. AVG now includes real-time security updates, scans for both malware and performance issues, and even catches malicious downloads before they reach your PC. User can Install AVG Antivirus on their computer to protect thier computer and network...
    www.avg.com/retail

    ReplyDelete
  88. Managing accounts is a tough task which is why hiring professionals is a great choice. Bookkeeping services of Alexilum are great. Our bookkeeping rates Brisbane are quite affordable.

    ReplyDelete
  89. QuickBooks is one such accounting software developed to help the bookkeepers and accountants in recording and reporting the financial operations of a company. It is an amazing system that is preferred all over the world by business firms. QuickBooks Support

    ReplyDelete
  90. Thanks for sharing this best stuff with us

    Black Horse Pest Control is an affordable and cheap pest control company. The service that we provide is worth the price that you pay. We will search for conditions that welcome pests and will stop it completely.

    ReplyDelete
  91. Search Engine Optimization takes the rank of your site on the next level. SEO is indeed essential since, without it, your website cannot survive. SEO works wonderfully to any location.
    Content Writing Company in Delhi
    Content Writing Services in Delhi
    Mobile App Development Company Delhi
    PPC Company in Delhi
    PPC Company in India

    ReplyDelete
  92. Fantastic blog... I think it's a very interesting and beneficial website for us.
    office.com/setup

    ReplyDelete
  93. Office.com/setup - Follow steps to download, install, and activate the Office setup. Click office.com/setup to enter the product key.

    ReplyDelete
  94. Norton.com/setup - Get esteemed protection on your system by downloading, installing, and activating the Norton setup. Visit norton.com/setup to get Norton.

    ReplyDelete
  95. Norton.com/setup-Get the Norton setup by downloading, installing on the Windows and Mac. Activate the Norton by entering the 25-digit alphanumeric product key at norton.com/setup.
    Visit : norton.com/setup

    ReplyDelete
  96. Download the MS Office setup on office.com/setup and easily activate your Office subscription by redeeming the activation key on office.com/setup.
    Visit: office.com/setup

    ReplyDelete