master-solving-logarithmic-equations-by-converting-to-exponential-equations

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 solve logarithmic equations by converting to exponential form. And the main basic thing of this is remember when we worked on converting from exponential to logarithmic form? Yes, it was very important for understanding what logarithms were, and how they're related to exponential. But for solving logarithmic equations, it's essential for us to understand how to convert them back into there, because here, I have a variable log base 3 of x equals 4. I need to solve for x. Well, I can't just type in that into my calculator and solve for the x. I got to be able to somehow find a way to isolate the variables. So I have to get x. You notice that x is inside of my function. Its log base 3 of x. So I got to get x outside of the function.

Well, by converting them rewriting my logarithmic equation in exponential form, I can rewrite this as 3 to the fourth power equals x. So now, to solve for x, all I simply need to do is take 3 to the fourth power. Well, you could plug in into your calculator, or you could also have been working on this for a while and understand that 3 to the fourth power is 81. So x equals 81.

OK, the next one again trips up a lot of students because they don't see the base, and they're like, what is the base for ln? Remember the base for ln is e. So again, I can still do the same thing. I'm just going to rewrite this in exponential form. So it's e to the third power equals x. However, I can't take e because e is an irrational number. So I am going to use my calculator for this one. So in my calculator, I'll just do second natural log, which will give me e raised to a power. And I'll just type in 3. And I get approximately-- x is approximately-- I'll round this to the nearest tenth. 20.1.

OK, so now we have some expressions. And that's what we're going to get into. Now are the main important thing I want to drill home with you on these types of problems is when we're solving a logarithmic equation by converting it to exponential form, we have to have the logarithm isolated, right? The logarithm cannot be adding or subtracting by the number, or multiplying or dividing by any number. In this case, you can see it might look like you're adding 25. But that's not the case. You're actually adding 25 inside the function. See, that's inside the parentheses. So that's OK. Now all I need to do is just convert it to exponential form because that's inside the function. It's not outside the function. The function is isolated.

So by rewriting this, I have 2 to the fourth equals x plus 25. Now when I take 2 to the fourth power, I get 16 equals x plus 25. And then I just solve for x. So I subtract 25 on both sides, and I get negative 9 equals x. X equals negative 9. Now I should actually state that we should always check for extraneous solutions. And what I mean by that is if you look at the graph of a logarithm, notice that the x values are negative numbers. So we can't take the logarithm of a negative number.

And a lot of times, this gets scary for people when they see their solution is negative. However, if I plug negative 9 back in for x, negative 9 plus 25 is 16. So I'm still taking the logarithm of a positive number. When I plug 20 in for x, I'm still taking the logarithm of a positive number. Plug 81 in for x, I'm still taking the logarithm of a positive number. So as long as I'm not taking the logarithm of a negative number, we're good. There's no extraneous solutions. And we'll get into that more next video.

All right. So in this case, you see I'm multiplying by 2 inside the function, but I'm also multiplying by 6 outside the function. So my function is not isolated. So to isolate it, what I need to do is divide by 6 on both sides. I don't need to divide by 2 because that's inside the function. So I have ln of 2x is equal to 5. Now I'll just do the exact same thing. Convert it to exponential form, then divide by 2 on both sides. So I have x is equal to e to the fifth divided by 2. So I do e raised to the fifth divided by 2. And I get approximately 74.2.

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