in this video I will go over the material covered by lecture 0.1.2 called solving equations and again just like before I wasn't going to cover anything here because uh solving equations is something that is extensively covered in pre-calculus or algebra or trigonometry this is where you learn how to solve all this equations however the most important thing here is not for um me to remind you of all the different ways how to solve all the different equations but to point out that in this class you are allowed to use technology to perform any precalculus task and solving equation does qualify as such and there are several great tools that I want you to not several great tools but there are at least more than one way of solving an equation with technology and the two main technology two main websites that I encourage you to use are the Desmos envelope from alpha here I give you um example of how to solve an equation um like using technology in two different ways one way is as simple as type the equation involved from Alpha and the other one could also be as simple as typing the equation in decimals but you can also be more um curious about the process of solving the equation and instead of just typing the equation you can actually break it up into uh steps such as Define a function for each side of the equation and then instead of just staring at some dots on the x-axis you will be actually looking at the intersections of two graphs you will still be looking at the X values because to solve an equation uh means to find the X values that make the equation a true statement but at least this provides more purpose to the process of solving the equation it becomes more visual well from alpha as you can see does it by default so you don't have to tell it while from up is actually a pretty cool tool so I would encourage everyone to start exploring both of this options obviously I do want you to read everything that I typed here it came from the bottom of my heart and I really really wanted to to share this with you and once you're done reading this and you're ready to do the homework let's Jump Right In and start solving equations with technology and I don't know how many homeworks we're gonna have but when you solve equation you solve equation it really doesn't matter whether it's logarithmic or trigonometric or exponential and speaking of devil here is the logarithmic equation let's solve it how do we solve it we're going to solve it with technology now it is possible to solve this one without technology and if you can do it for you um I'm gonna solve it with technology and let's go with the most um uh with the I guess with the most curious approach first so first of all I'm going to Define two functions uh one for each side so I'm going to define the function for the left hand side of the equation and of course we Define the function for the right side of the equation uh solving the equation now becomes as simple as finding the x value of the point of intersection so you can tell that these two intersect at x equals 8.624 8.624 however if I just type here x is equal to a point six two four it is not going to like it why because it is asking me for the exact value however in real life uh there isn't much I wasn't going to say there isn't much difference but there is a difference but from the Practical point of view sometimes 8.624 is just as good as the exact answer so when you go to buy lumber at the Home Depot and you tell them oh please could you cut me a square root of three uh feet of uh two by two um you will be looked at as a as a nut ball uh and again while there is a value in approximations in this particular case it does say to find the exact value so to find the exact value we are going and to just waste one attempt here to see that it's not what they want it just because it's not what they want doesn't mean it's the wrong answer so let's actually find that the answer so we're going to type negative three uh Ln five x minus 4 plus 4 equals negative seven in in both from alpha you can just type this it's just that easy it understands it notice that in Wolfram Alpha log is natural log so whenever you copy the answer from Wolfram Alpha make sure that you interpret blog as a natural log that is sln now notice how the graph that model produces pretty much the same wrap as we have right but Wolfram Alpha is definitely better at solving equations uh and providing the exact answer so this is the exact answer if you type this in a calculator so X is equal to one fifth uh e to the power supply I already forgot everything r e r eleven thirds plus four plus e to the power 11 thirds uh you see that that's exactly the same value that we're looking for and it's also 8.624 right however this is the answer that the software is looking for in this particular case but please understand that the fact that you're allowed to use technology means these problems can be done fast I don't need you to spend 40 minutes solving this logarithmic equation can you if you can can you have 30-40 minutes go for it but you don't have to however let me show you real quick how I would have solved this analytically if I had to and this is probably anyone would have done it this way we would have subtracted 4 from both sides we would have divided everything by negative 3 which would have eliminated this negative and then we would have raised both sides to the power we would exponentiate both sides so it would be e to the power 11 thirds and then we would have had four to both sides do you see where all those numbers are coming from and then we would have divided by five so this is how they got what they got but technology is just much faster and at this point we assume that you know how to solve these equations uh right this one let's do the same thing let's first use a Desmos and then we use walker Martha so Ln 2x minus 2 Ln x squared and on the right side we have zero so what we're looking for is the actual x-intercepts oh 1.26 I wonder if it's rounded or not well we can totally waste an attempt nope obviously we had to do something here but there's no arm pain trying different things uh let's see what open Office is oh wow too you would have thought let's check uh cube root of two I don't know how to type cube root of two but we know that cubicle of two is the same as 2 to the power uh one third and yeah that's that's very accurate that's pretty cool so I'm going to answer here 2 to the power I want to search yep and that's the right answer uh now group we solved it analytically I'm not going to but here's how foreign exponential equation same same approach I'm going to take uh Desmos first this one has been very picky now uh and clearly we see it has one solution right so we already know that much sometimes that's all you need to know uh we know approximately what it is but that's not what the software is looking for so let's just type this in move from Alpha and let's have it well from alpha does what I'm suggesting you to do right it graphs it uh it shows you it has one solution right away and then it gives you the the answer now this is clearly like a computer generated answer so you have to be careful when using technology if you write something like this on the test it it clearly means you use technology um but we can go well either from here or from here but either way the correct answer is uh I was trying to where is the log logarithm huh can I answered oh yeah so it's log base 6 of to on to 233 280. uh that's the right answer let's submit yep we got it um it's called the change of Base formula to go from here to here but uh also we can go from here this is exponential form of the answer and this is located in the form of the answer all right we're moving along and another exponential equation again doing it by hand I don't know how long it would have taken but uh but it's just not fun to do it by hand sometimes you just need to get things done and uh This is How we get things done uh this equation has two solutions right so when there are two solutions the instructions say to separate them by commas and uh what are the two solutions again it's probably going to have some radicals in them well this is exponential function right so we're probably going to have some natural log and and maybe of radicals but again what we learned right away from graphing it is that there are two solutions now if we want to actually find it both Solutions let's do this press enter oh look for it looks it's interesting uh the again remember log four log six that's in human language that's a land for comma Ln six other species uh I don't know let's see yep that's all there is now look at this analytical solution who has time for this not us another exponential equation same task so let's just get this done three e to the power negative negative five x minus a and minus four and nine so we see this equation has how many solutions uh one we know approximately what it is uh to find it we're going to use full form Alpha three e to the power negative five x minus a minus four uh equals nine and eventually you'll make a typo and you get the wrong answer and then you'll get confused why is this wrong I did everything right well just make sure you type everything right uh parentheses make sure every opening parenthesis has the closing parenthesis make sure you type a Ln or a log when it was appropriate in each situation and uh yeah but inevitably sooner or later you will get this wrong and then you'll have to redo it so it's really not a big deal there is no need to get frustrated with the with anything geez they just keep beating us up with this equation so um logarithmic equation again if you use just your brain and a piece of paper then you really need to remember how to solve algorithmic equations Olympic equations can get really ugly very fast uh just by you know adding another log in there and that's it you may not even be able to solve it analytically um so technology is the way to go again we can see that there is only one solution to find it I'm going to type the equation actually I'm curious if it's gonna understand me here but we'll see we'll see does it understand yes it does and where is the answer oh look at this what the answer is pretty nice I guess that's in calculus this is pretty nice I guess it may not look nice for an algebra student but for example student we do like this answer as well I hope this is it yay we are done is there another uh homework or nope that was it for lesson zero point uh one