 This video is going to talk about logs in an introduction and the main thing you need to remember is that logs are exponent when I was in high school and We were doing logs. We didn't have wonderful calculators That's how old I am and we had all these numbers in the back the book They were used this is so charted decimal numbers and to me That's all a log was was just these decimals and I really didn't know what a log was and so I started Teaching and I was trying to help my students learn about logs and I really had a hard time So I finally found a colleague that just brought it to light to me literally they just told me That remember the X logs are exponents and it was a light bulb to me is like that's what all those numbers were So logs are exponents and they are a number if you can keep that in mind. That'll make logs a little bit better Now when you see log that looks like this here, it's red log base B of X So the base is going to be Some little subscript number down here, and then you're going to have this X which is actually called an argument So why equal log base B? Means that why over here on this other side remember it logs an expert our exponent So why is the exponent on? B The base that will give you this X So we want to simplify these logs and remember it translates to find the exponent on the base that makes a statement true so if I have log base 2 of 8 I'm really saying 2 to the what is going to be equal to 8 and We all know that hopefully or you could always calculate it yourself 2 times 2 times 2 is 8 So 2 to the third log base 2 of 8 is equal to 3. That's the exponent on 2 So again over here log base 3 of 81. It's what exponent on 3 will give me 81 And if you don't know you could always use your calculator You come in here and say okay. Well three Karatex sometimes they give you a really big number that you don't know what happened to what it is Then we can go look at our table and let's look at some smaller numbers here Now if we look at our table and We're looking for 81 in this y column, and it looks like it's four Right here is four So we would say that log base three of 81 is four notice. I found that log by actually thinking about the exponential Let's look at this one five log base five of six twenty five so again not six five to the something it's going to be six twenty five well, I Know I know that five to the third is one twenty five So I would guess that five to the fourth is Going to be six twenty five, and I could always take one twenty five times Five this would be the long way to go, but carry the two. There's ten plus two Makes twelve five plus one more makes six so sure enough or five to the four is six twenty five That's the ways you can figure these out Sometimes you just know them sometimes you could use your calculator Sometimes you could use what you know to help you get to the next part Four squared is sixteen, and if I that's four squared During it the long way again if I multiply that by four I'm pretty sure that's going to be sixty four because six times four is twenty four carry the two Four plus two more would be sixty four, so I had two factors. I got one more So four to the third must be sixty four you can also express these like I Converting it log base b of x is equal to b to the y equal x I always say it's like hopping across the pond you take the base because we know exponential starts start with bases You hop across the pond to get the exponent Remember that's the exponent over here, and then you hop across The equal sign to get to the other side So you start with your base We hopped across to find our exponent, and then we hop back across to get to what it's equal to so to Hop across the pond gives us two to the seventh, and that's going to be equal to this x two to the seventh we could just go right to our calculator with that and Just go going to our home screen to care at seven. That's something we can compute and Two to the seventh is equal to one twenty eight So x must be one twenty eight now This is our answer right here, but the true statement would be Log base two of one twenty eight would be equal to seven right convert this one We start with our base. That's x hop across the pond to get the two and Then that's equal to hopping back across the pond Equal to thirty six Variable now is our base, and we can take the square root of both sides So the square root of x squared is going to be x and the square root of thirty six is going to be Six now some of you might be saying isn't that supposed to be a plus or minus six? Remember about bases the base has to be greater than zero and So we can only accept the x equals six. Let's try again Start with your base hop across the pond negative four Is equal to hopping back across the pond the x So five to the negative four is one over five to the fourth and I think earlier We found out that five to the fourth was six twenty five So using that fact now we have one over six twenty five Would be equal to x and That is a perfectly fine answer x Raised the third is Equal to the eight If you can just keep reminding yourselves that the other side of the equal sign is the exponent Then this conversion can be a little bit easier or maybe hopping across the pond helps. I don't know again My base now is my variable so I can take the cube root of both sides This gives me x on this side and the cube root of eight is two or if you needed the calculator you would say go to math cube root is four of eight is equal to two We can also Calculate some logs if they are common logs. That's base ten. That's what our number system is based on We got our ones or tens or hundreds or thousands. We're not base ten So that's why we could call it the common log because that's our number system So we have a log key because we use it so often and then base e is 2.718 blah blah blah blah It's a natural number. It's called the natural log and it's L n. That's not an I it's an L and in Latin this would be logarithmic naturalis or something like that. So that's why it's L n the L comes first in case you care and So we can just calculate these right from our calculator log 60 notice There's no a subscript down here because we know it's base 10 So it just log in our calculator and then 60 and log 60 is equal to I lost it approximately 1.7 and then we could do the same thing with the L n now the L n is below the log key and It's L n 3.5 And we get 1.25 approximately So now we know how to figure out logs if they're logs or natural logs, we can go to our calculator if they're Equations then we can convert them into an exponential and go from there Or if they're just asking us to evaluate a log We just have to think what is the exponent on our base that'll give us this number which again is called the argument