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Interactive video lesson plan for: Master Evaluating a logarithm without a calculator

Activity overview:

Subscribe! Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out. Welcome ladies and gentlemen. And what I'd like to do is show you how to evaluate logarithms without using a calculator. Now, obviously for a majority of these-- or actually for all of these, you could type them into your calculator using the change of base formula unless your calculator can do different bases. But none of these I chose base 10 or I didn't do any natural logarithms. But for all logarithms you can always just type in your calculator. But I guarantee I could do all of these as fast as it would take you to pick up your calculator and plug them in. And it's just because using the understanding of logarithms has helped me been able to do all these problems, as well as negative exponents.

So that's one thing I want to write into here is-- well, there's a couple things. First of all, remember we have a logarithms base b of x equals y. We can always rewrite that in logarithmic form as b to the y equals x. The other thing I want to remind you is if I have an exponent in the denominator, I can always rewrite that as an exponent in the numerator, but just using it with a negative power. You can put that over 1 if you wanted to but we don't really need to. So basically, by knowing these two understandings, I can do all these problems without a calculator very, very quickly.

The other thing, actually, we also want to know is using the one to one property is sometimes helpful even though we're not solving, we're just evaluating. A lot of these, it is very important to understand this because I think I'm going to do some difficult problems. So if we have an exponent equal to another exponent where the bases are exactly same, then their powers are same. So we can set their powers equal to each other. OK, now, you can see that these are just expressions that we want to evaluate. But our basic mode of thinking is, again, to find what their value is.

Now, remember when we're looking at a logarithmic equation-- for instance, log base 3 of 9-- we know that what that answer is going to be is 2. Because what a logarithm says and means is 2 raised to-- oh, I'm sorry, 3 raised to what power is 9? 2, and if you were to rewrite that in exponential form, it would look like this. 3 squared equals 9. So again, that's writing it into my exponential form. 3, the base, raised to what power gives you 9? That's what our logarithm is basically asking us. 3 raised to what power is 9? 2, and you can always think about rewriting it in exponential form to kind of check your answer.

So, in my first example I kind of tried to think of one that's going to be fairly simple. This is basically stating us 5 raised to what power is 25? Well, I know that 25 is a square number. And I know that 5 squared is 25. So therefore the answer here in this case, it's just going to be 2, OK. Now, the next one is 3 raised to what power is 81? Now again, without a calculator, you need to kind of start knowing the powers of 3, 4, 5, 6, 2. So what I would start doing, is if you're not really familiar what this would be, start raising 3 to powers. 3 to the first power is 3. 3 squared is 9. 3 cubed you might have to type in your calculator, but eventually you'll understand it's 27.

3 to the fourth power you might have to type in your calculator originally, but you'll see that it's 81. So that means 3 raised to what power is 81? Well, the answer is equal to 4, OK. Now, in the next example what I'm going to do is I'm just going to kind of think about it in another format. Now, you could do the exact same process I did over there, and just start listing 2 to what power, you know, equals that. But I also want you to think about this as maybe possibly an equation. Here's an expression. Now, again these are expressions. We're just evaluating. We're not equations. But really what an expression is is you're saying it's equal to what value?

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