WEBVTT
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So we want to find this on the solution to
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this equation. So we want to solve for X
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. Now we have that. The log, a
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rhythm of something involving X, is equal to three
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and two software X. We need toe isolate X
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. So recalled it. The natural law algorithm is
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log based. Eat. So it seems like we
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should take e to both sides. So this equation
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should be equivalent to e to the log of two
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X minus one equals e to the third. Okay
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, but e raised to the natural log, which
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is long base e. Remember, that natural log
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is really log based e cancels. And so we
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have two X minus one is equal to eat to
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the third. But now I could just use addition
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division, so I can add one. So two
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X is equal to e to the third plus one
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and then divide by two e to the third plus
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one, all divided by two, and this will
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be our final answer. So again, we're just
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using inverse relationship between logarithms and exponential is okay,
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so we have another equation and we want to software
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X. But this is actually going to illustrate something
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very interesting that can happen with equation solving. And
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we may see this from time to time. And
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it's something that we need to be very careful of
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when we're solving equations. But for now, let's
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just go in blind. Let's just say Okay,
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well, I want to solve this for X.
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So let me isolate X and I'm gonna be really
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clever, and I'm going to use properties of logarithms
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. I'm going to use the fact that if I'm
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adding to longer those with same base, I could
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just write them as a single longer them with arguments
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multiplied together. So I'm just writing this very clearly
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to say Okay, the log. I'm adding these
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two together, these air, all log based ease
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, I could just multiply the arguments. Well,
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that's great, because now I can just do my
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favorite trick and raised I could so take both sides
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and put him in the argument of e so e
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to this is equal to eat to this. That
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should be equivalent and then cancel the logs. I
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mean, if you if you think about it,
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you have long of something equals log of something else
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we're actually just using the fact that log is 1
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to 1. So if the logs of something are
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the same than the actually what you're taking, the
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log of has to be the same. So in
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other words, X plus six times X minus three
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is equal to 10. But this is just a
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quadratic equation. And now be careful. Don't use
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the zero product property here. Don't say either X
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plus six is 10 or X minus three is 10
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because it only works if a product of two things
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equal zero. So I actually need to expand this
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out. So let's do that. So this is
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X squared. We have minus three x plus six
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x that's plus three x minus 18. He goes
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10. I can subtract 10 over whenever you're solving
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a quadratic equation are really in the equation. It's
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always a good idea to get everything on one side
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, because now, when I factor notice that this
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does indeed factor, so we'll have X plus seven
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X minus four. I have a product of things
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equal to zero, and now I can use the
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zero product property so either X plus seven zero or
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X minus 40 And so we have two solutions.
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X equals negative seven and X equals four. But
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hold on, hold the phone. We're missing something
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very important. Now pause for a second and see
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if you could tell me what it iss. Of
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course you can't actually tell me, but just think
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about it. So one of these solutions is actually
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invalid. So if I plug in X is equal
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to four, I have log of 10 plus log
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of one. That's gonna be a log of 10
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plus zero. Log of+10 equals Log of five
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, plus log of two, which is long.
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10. So X equals four works, but try
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to plug in negative seven. Log of negative seven
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plus six is log of negative one. But the
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domain of the longer than function, remember, is
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on Lee positive numbers. I can Onley plug in
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positive numbers into a longer than so something is going
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on here. It doesn't even make sense to plug
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in negative seven into this equation. And the moral
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of the story is this. Always check. Check
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your answers. So when you find answers, I
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mean, I'm guilty of not doing this all the
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time, but it's always a good idea to go
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and check back and make sure that your solutions air
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actually solutions. Because in this case, this guy
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is not a solution. We Onley have one solution
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X is equal to four. So this guy right
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here sexually has a special name. This is called
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an extraneous solution. And an extraneous solution basically comes
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from doing something to the problem that actually wasn't 100%
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of valid. So actually, when we applied these
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properties of the logarithms, there's actually something we needed
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to make sure off, and I'm not going to
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go into a lot of detail. But essentially,
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I'm doing this step going from here to here.
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We actually introduced this extraneous solution and that's OK.
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As long as we check our answer and realized that
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Okay, X equals negative. Seven isn't actually a
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solution. Onley X is equal to four is a
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solution. So it's just again all of this stuff
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from pre calculus is just something to keep in the
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back of your mind because we're going to rapid fire
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. When we get to calculus, we're just gonna
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be doing algebra left and right, and all of
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these things are going to come to play from time
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to time