College Algebra: Logarithmic Exponents

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Let's solve some more of these equations, and just get into the habit of solving these things by using these very simple techniques, in order to untangle these really complicated-looking equations.
Now, here's one. This is one of those sheep-in-wolf's-clothing questions - e^ln(x-1) = 4. Now, there are two ways of answering this, two ways of solving for x. Step 1 is to be lazy, think, and say, "Wait a minute. What does this equal?" This right there - that is the exponent that I have to raise e to, in order to make it x - 1. Right? That is the exponent. That's the power I have to raise e to, in order to have it be x - 1. Look at what I'm doing - I'm taking e to that power. So, what does it equal? It equals x - 1, because I'm raising e to the very power I have to raise it to, in order to get x - 1. I could immediately just say, right from here, e raised to the natural log of blop - those two things cancel each other out, and I'm left with just x - 1. Therefore, x = 3.
You can check your answer by putting in 3, here. Let's see, 3 - 1 is 2, so what I see here is e - no, I made a mistake. Look what I'm doing here. I get so excited, because I got to an easy thing, look what I did. Do you see it? I brought this over, and it became a 3? Right. If you bring this over - I have to add.
Let me give a little lesson in addition. The lesson is the following: If you have 4 + 1, that actually equals 5. Now, here's a place where I made a mistake, but I'm not going to worry about it at all, because whenever I see a log, I always check my answer. I found that mistake when I was about to check, because it wasn't working out.
Let's check the answer now. If I put a 5 in there - 5 - 1 = 4, and e to the natural log of 4 is 4. That checks. So, in fact, an easy way to do this problem is just to realize that e raised to the natural log of blop is just blop." In fact, in general, if you have a base b, raised to the log base b of something, that always equals the something. They undo each other. They're actually inverse functions, so they cancel each other out, and we come up with A.
Now, suppose we didn't see that. Let me just show you that, in fact, this is really something we could answer, even if we weren't really slick like that. I could think, "Well, wait a minute. I've got the x's in the exponent, so I'm going to take the natural log of both sides." If I take the natural log of both sides, I'd see ln(e^ln(x-1)) = ln4.
Now, I'll use the property of exponents and logs that say if you have an exponent there, it becomes a coefficient. That's the whole reason why you take the natural log of both sides here - ln(x-1)lne = ln4. But what's the natural log of e? Remember that the natural log is log base e. So, this is log base e of e. That's what I have to raise e to, in order to get e. Well, e to what power, gives me e? 1. So, in fact, this is just 1. I can just drop that out, and I see ln(x-1) = ln4.
Now, what could you do? Well, since I have the natural log of something, equals the natural log of something else, those somethings must be equal. So, therefore, I see x -1 = 4, and then if you do it incorrectly, you'd say x = 3. If you do it correctly, you'd say x = 5. There's the answer. Got it before, in one step. Here, I did it in one, two, three, four steps. Either way, perfectly correct.
Let's try another funky one, because this is a log(log x) = 1. What do you do? Just follow the procedure that we already talked about and worked on together. When you see logs, untangle them. Log - remember invisible base 10 there, so what does this mean? It means that log is the exponent. So, 10 to the first power equals that. What this says is 10 to the first power equals that - log x. Now, what do you undo? Untangle again, keep on untangling. This is invisible 10 here, and so, what do I see? Log is the exponent - 10 is the exponent I have to raise 10 to, in order to get x. So, x = 10^10. That's a big number. That's a 1 with 10 zeros.
Again, you can check your answer, and you should check your answer, because you have logs there, by plugging this in. What's log(log 10^10)? Well, think about that. What do I do? Well, first of all, this little 10 exponent can be pulled out in front of this. That equals log of - and then I pull that 10 out, and I see (10 log 10). But, what's log of 10? Well, remember, log of 10 is log base 10 of 10. That's the exponent I have to raise 10 to, in order to get 10. So, 10 to what power equals 10? Well, 1. This is actually just 1, so I have natural log of 10 x 1. What's the natural log of 10? Well, that's still 1, and look - we get the 1. This checks. So, the answer is the big answer of 10^10 - pretty big.
The last one I want to take a look at is ln(e^3x) = 6. Well, I don't even have to worry about taking logs here, because I have log of something here. This immediately comes out in front, without any labor at all. I see 3xlne = 6. What's natural log of e? That's log base e of e, which is 1. It's 1 - natural log of e is 1. I just see 3x = 6, so x = 2. You can check that answer by putting this back in, and I see ln(e^6). Does that equal 6? Well, I bring this out in front, and I see 6lne. The natural log of e is 1, so this is a 6. So, in fact, it does check.
Exponential and Logarithmic Functions
Solving Exponential and Logarithmic Equations
Solving Equations with Logarithmic Exponents Page [2 of 2]