Master Expanding Logarithmic Expressions using the rules of logarithms

Subscribe! http://www.freemathvideos.com 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. So what I'd like to do is show you how to expand logarithmic expressions. Now expanding logarithmic expressions, what we're going to do is follow the rules of logarithms. And the rules of logarithms we have product rule, quotient rule, as well as power rule. And I've put them up here, here's the product rule, here's the quotient rule, here's the power rule. And then I wrote two other little rules of exponents that could come in handy as far as rational powers. You can rewrite as a radical, and vice versa, negative powers you can rewrite in the denominator.

So there's multiple different ways to do this. I'm just going to do it the way that I think would be best. But just so you know, there's different ways for some of these problems. We'll still arrive at the exact same simplified expression or expanded expression.

So basically what we have when we're expanding is we're given one single logarithmic expression, and what we want to do is expand it. And we're going to basically be using these three properties of logarithms to do just that. So basically what we want to do is first kind of identify what type of property that we have. So in this case, I have log base 3 of 5x. Well 5x is multiplication. Right? You're multiplying the 5 and the x. Well what the properties of logarithm state is if you have the logarithm of a base b of the product of two terms, what we can do is rewrite that as two separate logarithms with that same base, by adding the two logarithms. So this one is just going to be log base 3 of 5, plus log base 3 of x.

The next example is going to be using the quotient property, you can see that I am dividing my x and y. So just like I'm dividing my m and the n , I'm now going to rewrite that as subtraction. And again it doesn't matter, I use the properties of logarithms for base 10, or for any regular logarithms, but you it works the same for natural logarithms as well as a logarithm for any base. So therefore this is going to be the ln of x minus ln of y.

The last major property is the power rule, and basically what the power rule of logarithms states is if you have a logarithm of base b of m, raise to the n power, we can rewrite that as the product of that power times your logarithm. So we can take 4 and rewrite it in front and write log x. Now, the rules of logarithms are not just-- I just did one property a time. But now what we're going to do is start getting into logs simplifying or expanding logarithms where there's going to be multiple properties involved.

On this next one it might not be apparent on what exactly I can do where the radical come from right that's not available in there well by using our rules of exponents we understand that a radical can be rewritten as a power. So I can rewrite this as log base 2 of y to the 1/3. Now I can take that 1/3 and put it in front. OK?

In the next one you can see now I have three different logarithms. Well with each logarithm I am multiplying. So now all I'm going to do is apply the product rule, not once but twice. I need to separate these two as multiplication, and these two. Now, you can do it in certain steps, right? You can do one at a time, or you can just break up each one of these. So now I'm going to do log of y squared plus log of x to the negative 1/3 plus log of the square root of z. OK?

Now the main important thing here is you can see that now I have powers in each one, well not in each one of those. I can rewrite this as raise to the 1/2. Right? Well now you can see each of my logarithms have a power. Well we don't like having powers, right? We want to write those powers in front. So to write it in fully expanded form I'm going to bring the 2 down, I'm going to bring down the negative 3, and I'm going to bring down the 1/2. And usually plus or minus we just write it as minus. OK? But there you go, that would be our fully expanded form.

Видео Master Expanding Logarithmic Expressions using the rules of logarithms канала Brian McLogan