Video transcript
– [ Voiceover ] Let ‘s get some practice solving some exponential equations, and we have one correct over here. We have 26 to the 9x plus five ability equals one. so, pause the television 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 baron, to the zeroth power is equal to one. Anything to the zeroth power is going to be peer 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. indeed, we fair have to say, well, 9x plus five needs to be peer 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 damaging 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 moment more, a little sting more concern. Let ‘s say we have the exponential equation two to the 3x plus five power is adequate to 64 to the x minus seventh world power. once again, pause the video, and see if you can tell me what ten 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 hour, possibly 3x plus five needs to be equal to x minus seven, but that would n’t work, because these are two different bases. You have two to the 3x plus five world power, and then you have 64 to the x minus seven. so, the key here is to express both of these with the lapp base, and golden for us, 64 is a power of two, two to the, let ‘s see, two to the third base is eight, so it ‘s going to be two to the third base times two to the third gear. Eight times eight is 64, so it ‘s two to the sixth is peer to 64, and you can verify that. Take six twos and multiply them together, you ‘re going to get 64. This was equitable a little spot easier for me. Eight times eight, and this is the lapp thing as two to the sixth might, is 64, and I knew it was two to the sixth ability because I precisely added the exponents because I had the same basal. All right, so I can rewrite 64. Let me rewrite the whole matter. therefore, this is two to the 3x plus five world power is peer to, alternatively of writing 64, I ‘m going to write two to the sixth might, two to the one-sixth exponent, and then that to the x minus seventh baron, x minus seven office. And to simplify this a small bit, we precisely have to remind ourselves that, if I raise something to one ability, and then I raise that to another power, this is the same thing as raising my base to the intersection of these powers, a to the bc power. so, this equation I can rewrite as two to the 3x plus five is adequate to two to the, and I just multiply six times ten minus 7, so it ‘s going to be 6x, 6x minus six times seven is 42. I ‘ll just write the solid thing in scandalmongering. so, 6x subtraction 42, I merely multiplied the six times the integral expression x minus 7. And so now it ‘s interest. I have two to the 3x plus 5 ability has to be equal to two to the 6x minus 42 office. So these necessitate to be the lapp advocate. indeed, 3x plus five needs to be equal to 6x minus 42. so there we go. It sets up a dainty 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 x 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 lease ‘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. ten is equal to 47 over three. And we are done.
