Hi guys, this lesson is on natural logarithms. So before we start using the natural log, let's talk about what it is. A natural log is a log base e. Okay, so this e right here, when we have that as a base for a logarithm, it has a special notation.
It's a natural log. That's the ln. Well, what is e? e is the natural base, and it is this number over here.
Okay, e is just a mathematical constant, similar to how pi is a mathematical constant. So for the first example, you're just evaluating using the calculator. So you can round to four decimal places. So you're taking that and plugging it in. So e to the 0.5 gives us 1.6487.
For letter b, I have e to the negative 2.67 gives me 0.0693 to four decimal places. Natural log, hitting the natural log on the calculator of 3, that gives me 1. point zero nine eight six and evaluating the natural log of one fourth gives me negative one point three eight six three. So the first example is just getting you used to where those buttons are on the calculator.
For the second example we're just switching back and forth between exponential and logarithmic form. So if you look at letter a, this is exponential form since it's written with x as an exponent. So your base for the exponential function is the same thing as the base for your log. So this is a log base e of 23. A logarithm is an exponent, so it always has to equal the exponent, which in this case is x.
Log base e now has a special notation, that's the natural log. So I can write this as the natural log of 23 equals x. For this particular example, you are not evaluating these, you're just switching back and forth between the forms, so your answer is just going to stay as natural log of 23. For letter B, we're starting with logarithmic form and changing it to exponential. So natural log is the same thing as a log base e, and the squiggly equal sign means approximately equal to.
So I have log base e of x is approximately equal to 1.2528. I'm going to change this to exponential form. So you're starting with your base. Swing across, there's your exponent.
Swing back across, and that's what it equals. So I have e to the 1.2528 equals x. And again, you're not evaluating these, so you don't have to take this and plug it in. You're just changing the form.
The inverse property of base e and natural logarithms. This goes back to the inverse property of regular logs. If your base for your exponential function is the same thing as the base for your log, so e with a natural log, it's going to cancel itself out.
So your final answer for this is just 21. And then same thing for letter b. I have a natural log, and then the base e, that's going to cancel itself out. So that gives you x squared minus 1. The inverse property is very important once we get to solving equations, which is what we're doing right now. Okay, so the first two scenarios, I have a natural log, and they equal a number.
So when a logarithm equals a number, what you want to do with that is you want to change it to exponential form. So this is a log base e. of 3x equals 0.5.
So I'm starting with my base. I'm going to swing across. There's my exponent. Swing back across, and that's what it equals. So I have e to the 0.5 equals 3x.
Well, e to the 0.5, we can go ahead and evaluate. That's 1.6487 equals 3x. Divide both sides by 3. And you get x equals 1.6487 divided by 3 gives you 0.5496. Same thing for letter b. So natural log is a log base e of 2x minus 3 equals 2.5.
So I start with my base for my logarithm, swing across. Swing back across. So I have e to the 2.5 equals 2x minus 3. e to the 2.5 is 12.1825 equals 2x minus 3. Add 3 to both sides. That gives you 15.1825 equals 2x.
Divide by 2, and you get x equals 7.5912. So these two problems, we had a logarithm equals a number, so we changed it to exponential form. For these last two problems, we are now going back to exponential equations. So exponential equations, you always need to start by isolating your exponential function.
This right here is your exponential function. So your base is the e. And you have your exponent. That's how you know it's the exponential function.
So we want to isolate it, so I'm going to start by subtracting 4 from both sides. That gives me 3e to the negative 2x equals 6. Divide by 3. e to the negative 2x equals 2. And since I'm solving for an exponential function, I'm going to incorporate logarithms. So in this case, since I'm dealing with a base e, I want to use the natural log. So I'm going to take the natural log of both sides. Now using the inverse property, the natural log of e is going to cancel itself out.
So you are left with negative 2x, each equals natural log of 2. So natural log of 2, I can plug that into the calculator. That gives me 0.6931. Divide by negative 2, that gives me negative 0.3466. And there is your answer.
You're going to do the same thing with letter D. I'm going to start by adding 5 to both sides. So that gives me 2e to the x plus 5 equals 14. Divide by 2. I have e to the x plus 5 equals 7. Again, I'm going to take the natural log of both sides. And I'm using the natural log instead of a regular log because natural logs and e's will always cancel out. So it's going to save us a step.
So natural log and e's going to cancel. So I have x plus 5 equals natural log of 7. Natural log of 7 I can plug in. That gives me 1.9459. Subtract 5 from both sides, and that gives me negative 3.0541. Alright, so that is everything for the lesson on natural logarithms.