{"page":"\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/resources-572d6a42.css\" /\u003e\n\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/lp_boclips_stylesheets-f4d0de30.css\" media=\"all\" /\u003e\n\u003cdiv data-title='Master Evaluating Logarithmic Expressions using the Change of Base Formula' data-url='/boclips/videos/6367368524a5c53148750be6' data-video-url='/boclips/videos/6367368524a5c53148750be6' id='bo_player_modal'\u003e\n\u003cdiv class='boclips-resource-page modal-dialog panel-container'\u003e\n\u003cdiv class='react-notifications-root'\u003e\u003c/div\u003e\n\u003cdiv class='rp-header'\u003e\n\u003cdiv class='rp-type'\u003e\n\u003ci aria-hidden='true' class='fai fa-regular fa-circle-play'\u003e\u003c/i\u003e\nVideo\n\u003c/div\u003e\n\u003ch1 class='rp-title' id='video-title'\u003e\nMaster Evaluating Logarithmic Expressions using the Change of Base Formula\n\u003c/h1\u003e\n\u003cdiv class='rp-actions'\u003e\n\u003cdiv class='mr-1'\u003e\n\u003ca class=\"btn btn-success\" data-posthog-event=\"Signup: LP Signup Activity\" data-posthog-location=\"body_link_boclips\" data-remote=\"true\" href=\"/subscription/new\"\u003e\u003cspan\u003e\u003cspan\u003eGet Free Access\u003c/span\u003e\u003cspan class=\"\"\u003e for 10 Days\u003c/span\u003e\u003cspan\u003e!\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-body'\u003e\n\u003cdiv class='rp-info'\u003e\n\u003cdiv aria-label='Hide resource details' class='rp-hide-info' role='button' tabindex='0'\u003e\u0026times;\u003c/div\u003e\n\u003ci aria-label='Expand resource details' class='rp-expand-info fai fa-solid fa-up-right-and-down-left-from-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003ci aria-label='Compress resource details' class='rp-compress-info fai fa-solid fa-down-left-and-up-right-to-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003cdiv class='rp-rating'\u003e\n\u003cspan class='resource-pool'\u003e\n\u003cspan class='pool-label'\u003ePublisher:\u003c/span\u003e\n\u003cspan class='pool-name'\u003e\n\u003cspan class='text'\u003e\u003ca data-publisher-id=\"30355462\" href=\"/search?publisher_ids%5B%5D=30355462\"\u003eBrian McLogan\u003c/a\u003e\u003c/span\u003e\n\u003c/span\u003e\n\u003c/span\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-description'\u003e\n\u003cspan class='short-description'\u003eWelcome, ladies and gentlemen. So what I'd like to do is show you how to use the change of base formula to evaluate for a logarithm. Now, while many calculators allow you to kind of choose whatever base you like to, some calculators you...\u003c/span\u003e\n\u003cspan class='full-description hide'\u003eWelcome, ladies and gentlemen. So what I'd like to do is show you how to use the change of base formula to evaluate for a logarithm. Now, while many calculators allow you to kind of choose whatever base you like to, some calculators you still can only use logarithms in base 10, and natural logarithms obviously would just be with base e. \u003cbr/\u003e\u003cbr/\u003eSo what I want to do is at least present the formula for you. And then I was also showing you, because if you have a calculator where you cannot determine the base, you have to use log base 10. So I'll kind of give you a couple different ways we can do this. \u003cbr/\u003e\u003cbr/\u003eNow for the first two problems though, you should at least know what the answers are. But I want to use the change of base formula so therefore we can see why it works and how it works. So basically the change of base formula is if you have a logarithm base b of a, then we can take the logarithm of any base y of a and divide it by a log of that same base of b. Or we could do the natural log, which would obviously therefore then if you did the natural log, then y would equal e, right? So it would be the natural log of a over the natural log of b. \u003cbr/\u003e\u003cbr/\u003eOr you could also just look at it as log a over log b in your calculator, because the majority of calculators-- for instance, like a scientific like this-- only has log and base 10. Whereas some TI calculators, you can actually change your base. \u003cbr/\u003e\u003cbr/\u003eSo if we were going to look at this, we would know this is going to be base 10, right? So what I want to do is let's go ahead and use this as log base 10. Let's do our change of base formula for this. \u003cbr/\u003e\u003cbr/\u003eSo therefore I can rewrite this as log base 10 of 100 divided by log base 10 of 10. Excuse me. So if I change this into log base 10 of 10 divided by log base 10 of 10. So that was my a, and this is my b. \u003cbr/\u003e\u003cbr/\u003eNow again, we already know what log base 10 of 10 is, right? 10 raised to what power gives you 100? We should know the answer is 2. Well 2 over 1, which just equals 2. \u003cbr/\u003e\u003cbr/\u003eYou know what, I don't want to do this again. Let's change this. Let's do this to-- let's do 4 over 8. That will work. OK, so now if I was going to a problem like this, log base 4 of 8. \u003cbr/\u003e\u003cbr/\u003eNow I can rewrite this to any logarithm, right? And the easiest probably would be, if you were going to use your calculator, would just be log of 8 divided by log of 4. And you can just plug it in your calculator and I'll show you what to do for those in just a second. \u003cbr/\u003e\u003cbr/\u003eHowever, I wanted to prove to you that this works. So if you were to look at this, instead of using log base 10, let's use log base 2. So therefore to rewrite this, I'm going to have my y is going to be 2. So my a is 8. So it's log base 2 of 8 over log base 2 of 4. \u003cbr/\u003e\u003cbr/\u003eNow remember, again, you can choose whatever base you want to. So your a and your b have to remain there, but you can choose whatever base you want to. For your calculator, you're going to have to use, more than likely, you're going to use log base 10, because you just type in log and it automatically will be base 10. Or you can use natural log. \u003cbr/\u003e\u003cbr/\u003eHowever, without a calculator, 2 raised to the eighth-- 2 raised to what power gives you 8-- 3. 2 raised to what power gives you 4-- 2. So the answer is 3/2. \u003cbr/\u003e\u003cbr/\u003eAnd if you were to verify that in your calculator, you could do log of 8. Actually, that's just do that. Log of 8 divided by log of 4. And what do you get? I get 1.5, which is 3/2, decimal version. \u003cbr/\u003e\u003cbr/\u003eAll right, so over here, again as I was mentioning, I'm going to work on this a couple different ways. So if I was going to use my calculator-- now in this case, I don't have the bases are the same. I can't rewrite them like I did over here and pick a base. So therefore I'm simply, for my calculator, I'm going to do log of 8 over the log of 3. \u003cbr/\u003e\u003cbr/\u003eSo simply in my calculator, I'm just going to find my log. And I'm going to do log of 8. I'm going to close my parentheses, divided by the log of 3, close parentheses. And I get 1.892789. So therefore I'm approximating. So they can say it's going to be approximately one point-- I'm going around to the 100th-- 89. 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