Topic B: Lessons 7-9: Logarithms intro
- [Voiceover] Let's get some practice solving some exponential equations, and we have one right over here. We have 26 to the 9x plus five power equals one. So, pause the video and see if you can tell me what x is going to be. Well, the key here is to realize that 26 to the zeroth power, to the zeroth power is equal to one. Anything to the zeroth power is going to be equal to one. Zero to the zeroth power we can discuss at some other time, but anything other than zero to the zeroth power is going to be one. So, we just have to say, well, 9x plus five needs to be equal to zero. 9x plus five needs to be equal to zero. And this is pretty straightforward to solve. Subtract five from both sides. And we get 9x is equal to negative five. Divide both sides by nine, and we are left with x is equal to negative five. Let's do another one of these, and let's make it a little bit more, a little bit more interesting. Let's say we have the exponential equation two to the 3x plus five power is equal to 64 to the x minus seventh power. Once again, pause the video, and see if you can tell me what x is going to be, or what x needs to be to satisfy this exponential equation. All right, so you might, at first, say, oh, wait a minute, maybe 3x plus five needs to be equal to x minus seven, but that wouldn't work, because these are two different bases. You have two to the 3x plus five power, and then you have 64 to the x minus seven. So, the key here is to express both of these with the same base, and lucky for us, 64 is a power of two, two to the, let's see, two to the third is eight, so it's going to be two to the third times two to the third. Eight times eight is 64, so it's two to the sixth is equal to 64, and you can verify that. Take six twos and multiply them together, you're going to get 64. This was just a little bit easier for me. Eight times eight, and this is the same thing as two to the sixth power, is 64, and I knew it was two to the sixth power because I just added the exponents because I had the same base. All right, so I can rewrite 64. Let me rewrite the whole thing. So, this is two to the 3x plus five power is equal to, instead of writing 64, I'm going to write two to the sixth power, two to the sixth power, and then that to the x minus seventh power, x minus seven power. And to simplify this a little bit, we just have to remind ourselves that, if I raise something to one power, and then I raise that to another power, this is the same thing as raising my base to the product of these powers, a to the bc power. So, this equation I can rewrite as two to the 3x plus five is equal to two to the, and I just multiply six times x minus 7, so it's going to be 6x, 6x minus six times seven is 42. I'll just write the whole thing in yellow. So, 6x minus 42, I just multiplied the six times the entire expression x minus 7. And so now it's interesting. I have two to the 3x plus 5 power has to be equal to two to the 6x minus 42 power. So these need to be the same exponent. So, 3x plus five needs to be equal to 6x minus 42. So there we go. It sets up a nice little linear equation for us. 3x plus five is equal to 6x minus 42. Let's see, we could get all of our, since, I'll put all my Xs on the right hand side, since I have more Xs on the right already, so let me subtract 3x from both sides. And let me, I want to get rid of this 42 here, so let's add 42 to both sides, and we are going to be left with five plus 42 is 47, is equal to, 47 is equal to 3x. Now we just divide both sides by three, and we are left with x is equal to 47 over three. X is equal to 47 over three. And we are done.