All right, we're going to talk about logarithms now. And what we're going to cover is examples of logarithms, what we mean by the characteristic in the mantissa of a logarithm, negative logarithms, anti logarithms, and at the very end natural logarithms. So let's set clear off this slide here and go to the next slide. Alright, so in this slide here, what is a logarithm? Well, we're talking about common logs here in this slide, we're talking about common logs. And and we talked about the powers of 10.

So if I look at examples of logarithm Okay, 10 to the one All right means multiply 10 by one, R. Okay? So 10 or one is the log of 10. All right, let's go to the next 110 squared or 10 to the to 100 or two is the log of 100. Okay? And 1003 because if I say 310 to the third we know that's 1000. So the log of 1000 is three.

The log of 10,000 is four. All right? So we can say that what we mean by a log, and again, this is for common logs. We're talking about common logs here. All right. The X ponent is the log of the number.

All right? And now if we go on the other side, and we're looking at the log of numbers less than one like point one, zero dot 010 dot 01, these guys here, okay, notice from our lesson on negative powers of 10. Okay, my exponent will be a minus number here. All right? So we can say that in common logs when we're asking for the exponent, I mean, we're when we're asking for a common law, the common log is the exponent used in the power of 10. May it be positive or negative.

All right. So let's stop here and see what's going on. In the next slot, okay, so let's, as we said in the previous slide logarithms are exponents. All right? An example here is the example of 10 to the two equal 100. The exponent is two.

So two is the logarithm. Okay? So logarithms to the base 10, again are called common logs. So let's give the following values Okay, so what would be 100? Well, it would be two. Now we can also use the calculated define common logs which I'm going to show you right now.

All right. Okay, well I brought down my calculated it's, it's with this operating system, and we're in scientific mode. And if you look, what does that say that says log right there. Okay. So we know Just by my previous explanation, and even what here the log of 100 is, too. So, I mean, I can look at that, no, but when logs get a little bit more complicated, as we'll see in a slide or two, it may be difficult to figure it out.

In the old days, we use something called the log table, but with the advent of the calculator that's kind of gone. So if I want to find the log of 100, I just plug in 100. And I click the Log key, and it's two. That's it. So we know the log of 100 is two. All right, so what's the log of zero dot 01?

Well, let's clear it and go zero dot 01. And let's click the Log key. That's a minus two, because I'm less than one. All right. So what What I'm gonna do now is I'm going to stop this. Okay?

Use the calculator. If you're on your computer, use the calculator that comes with the operating system. If not, I'm sure there's some battery powered calculators, just make sure you get one that says scientific. And you should have a long key there. All right, so we're going to stop the slide here. And when we go on, on the next slide, you'll see the, you'll see the answers and as always as a contact number.

And a way to get ahold of me via phone or via email on this platform. And we'll help you out as always. All right, let's stop this now and let you do the problems. All right, here are the answers. By doing the calculator they, they should be somewhat easy here. I'm going to pull my calculator down, and we're just gonna we're going to do a couple just to make sure you have You know, let me move that.

Let me move my calculator over here. All right, again, zero dot 001. log minus three is my answer right there. Okay, let's clear this. Let's do that log of 10,000. Hit the log key is a four. Let's clear it 100,000.

Log key five and the last one, 1 million. Long key six. All right, so we got them. All right, using the calculator and think about it. What if I got a number of 80? What if I got a number of 250?

Next slide. See you there. All right, as we start Before logarithms are exponents, and again I give you another example. Example 10 to the two equal 100. The exponent is two we talked about that again, logarithms are based on the called common logs, they are based on the base 10 numbering system. And any Here we go, any number can can represent by powers of 10.

All right, any number can be represented by the powers of 10. All right, exponent can be a fractional value. And that's what I said on the previous slide. What if I've got a number of 80? What if I got a number of 250? All right, but on this example here, right there it says therefore the log of 110 is one and the log of 100 is two Therefore the log of 63.

And even though I say it's 1.8 here, okay? The log of 63 has to be between 10 to the one and 10 squared. Okay? All right. So therefore we look at it, we use the calculator, which I'm going to demonstrate in a minute. Okay, if we find the log of 63 is 10 to the 1.8 power, like, again, previously, we had log charts.

And we're the next slide, we're going to talk about the mantissa and the characteristic. So what you would do is, and we'll see it there, you would find the characteristic, which is actually the number of decimal points, and then you would find the mantissa. And then you could figure out the power 10 to the power of whatever the exponent would be some fractional value. All right. Again, with the calculator, it's become so much easier. And I mean, you could probably find logarithm charts on the internet, but I mean calculator has just made it very, very nice.

Okay, so let me stop here, bring down my calculator, and we'll see how to do that. Okay, I brought down my calculator again and make sure you're in scientific mode. Once again, I'm using the calculator that's with this operating system. I'm using Windows. And when Well, what am I asking you here? The log of 63.

And again, from my example, we know it's going to be between one and two. Actually, it's going to be greater than one, but less than two. So all I do is I go in and 63 plug it in here. And I clicked the log key 1.799 Three? Well, we'll go on to two decimal points. So I rounded it up here to 1.8.

So what we're saying is 10 to the 1.8 equals 63. Right, approximately. All right? We're going to talk about anti logs. But if I wanted to check that, let me clear this. See where I got, I tend to the x.

So I would plug in 1.8. Right, and let's see what 10 to the 1.8 power is, because when I plugged the 1.8 in, this represents the little X up there. So let's hit that. And look at that it's 63 dot 095. I rounded it off to 63. So we're right.

The log of 63 is 1.8. Because if I raise 10 to the one point a power, I get six All right, as in the previous slide, I said we are going to talk about the characteristic and the mantissa. And that's exactly what we're going to do here on this slide. And again, I'm taking this exponent 1.8. And the one of the 1.8 is the characteristic. And the point eight is called the mantissa.

All right, so we have two parts of the logarithm, the characteristic and the mantissa. If you notice that back back when we talked about this, like, like the log of the log of 100 100 was two. So we didn't show it, but it was two and that was the log of 100. And it was too Well, that was my characteristic. Right there is my mantissa. All right, because why we had zero as the mantissa.

Did, we didn't need to show it. All right. But if I have a a number, that is more than in this example, that is more than one but less than 10, I'll have a fractional part. And again, that's called the mantissa. Then I show it and I have two parts. I let's clear the slide off here before it gets up.

Gets to miss messy. All right. So the cat again, the characteristic donate how many decimal points are in the number. The fractional part, again, is called the mantissa. All right, and then we put some numbers, I give you some examples, and this is what we're trying to do. We'll make a little bit of sense out of this.

Now, the numbers 2030 4050 6070 8090 all have a characteristic of one. Alright, you'll see why the mantissa will change with the number and I showed you an example here. So if we had got looking at 20, the law This is the law of 20. Okay, I don't know if I made it clear enough. So the log of 20, the characteristic is one, and the mantissa is three. So again, the log of 20 is 01.

Point three. If we go to another number that's greater than 20 and I just picked 60 for no reason. Okay, the characteristic still SES as one and the mantissa increases to 78. So therefore, the log of 60 is one dot seven, eight. All right, think about what we're doing again, and maybe I'm beating this too much. But the numbers are 20 3040 5060 7080 and 90.

All right, we know. We know that 10 to the one is 10 and 10 squared is 100. So we know that 2030 4050 6070 8090 has to be in between there. So they're going to be greater than my exponent one, but less than my exponent two. Okay. Let's stop here.

And let me clean up the slide well brought down my calculator, and I'm trying to lift it up there and you'll see why in a minute. All right, that's good. Okay, so now, again, let's put, let's find the log of 20. And it's 1.30. And we've got a decimal point I rounded it off to one decimal point. So there it is there.

And let's clear that. And let's plug in 60. And let's find the log of 60. And that's 1.778. I went to Duke decimal points here, and I got 1.78. So what does that mean?

Well, it means and let's take this one here. If I plug in 1.78 whoops, let me clear it up. But the decimal point 1.78 Go right there, which is 10 to the 1.78. I get 62. And I'm only showing 60 here. And that's because this the reason we're off a bit is because the log was a repeating decimal it went went up went to infinity.

So but we're good. Okay, we're good. All right. So that that said here so let's go on to the next slide.