0:00:00.590,0:00:05.690 We're asked to graph, y is[br]equal to log base 5 of x. 0:00:05.690,0:00:07.780 And just to remind us[br]what this is saying, 0:00:07.780,0:00:10.240 this is saying that y[br]is equal to the power 0:00:10.240,0:00:13.460 that I have to raise[br]5 to to get to x. 0:00:13.460,0:00:15.980 Or if I were to write[br]this logarithmic equation 0:00:15.980,0:00:19.510 as an exponential[br]equation, 5 is my base, 0:00:19.510,0:00:23.640 y is the exponent that I[br]have to raise my base to, 0:00:23.640,0:00:28.420 and then x is what I get when[br]I raise 5 to the yth power. 0:00:28.420,0:00:32.570 So another way of writing[br]this equation would be 5 0:00:32.570,0:00:40.710 to the y'th power is[br]going to be equal to x. 0:00:40.710,0:00:43.010 These are the same thing. 0:00:43.010,0:00:45.480 Here, we have y as[br]a function of x. 0:00:45.480,0:00:48.550 Here, we have x as[br]a function of y. 0:00:48.550,0:00:50.980 But they're really saying[br]the exact same thing, 0:00:50.980,0:00:53.624 raise 5 to the y'th[br]power to get x. 0:00:53.624,0:00:55.790 When you put it as a[br]logarithm, you're saying, well, 0:00:55.790,0:00:58.010 what power do I have[br]to raise 5 to to get x? 0:00:58.010,0:00:59.440 We'll have to raise it to y. 0:00:59.440,0:01:02.940 Here, what do I get when I[br]raise five to the y power? 0:01:02.940,0:01:04.099 I get x. 0:01:04.099,0:01:07.200 That out of the way, let's[br]make ourselves a little table 0:01:07.200,0:01:08.710 that we can use to[br]plot some points, 0:01:08.710,0:01:10.335 and then we can[br]connect the dots to see 0:01:10.335,0:01:11.980 what this curve looks like. 0:01:11.980,0:01:13.940 So let me pick some[br]x's and some y's. 0:01:18.500,0:01:21.100 And we, in general, want to[br]pick some numbers that give us 0:01:21.100,0:01:24.689 some nice round answers, some[br]nice fairly simple numbers 0:01:24.689,0:01:26.230 for us to deal with,[br]so that we don't 0:01:26.230,0:01:27.760 have to get the calculator. 0:01:27.760,0:01:29.590 And so in general,[br]you want to pick 0:01:29.590,0:01:34.250 x values where the power[br]that you have to raise 5 0:01:34.250,0:01:38.156 to to get that x value is a[br]pretty straightforward power. 0:01:38.156,0:01:39.530 Or another way to[br]think about it, 0:01:39.530,0:01:41.810 you could just think about[br]the different y values 0:01:41.810,0:01:44.740 that you want to raise[br]5 to the power of, 0:01:44.740,0:01:46.360 and then you could[br]get your x values. 0:01:46.360,0:01:48.590 So we could actually[br]think about this one 0:01:48.590,0:01:52.510 to come up with our[br]actual x values. 0:01:52.510,0:01:56.370 But we want to be clear that[br]when we express it like this, 0:01:56.370,0:01:59.850 the independent variable is x,[br]and the dependent variable is 0:01:59.850,0:02:00.480 y. 0:02:00.480,0:02:03.660 We might just look at[br]this one to pick some nice 0:02:03.660,0:02:10.440 even or nice x's that give[br]us nice clean answers for y. 0:02:10.440,0:02:12.690 So here, I'm actually going[br]to fill in the y first, 0:02:12.690,0:02:14.820 just so we get nice clean x's. 0:02:14.820,0:02:16.500 So let's say we're[br]going to raise five 0:02:16.500,0:02:19.120 to the-- let's say we're going[br]to raise it-- I'm going to pick 0:02:19.120,0:02:23.820 some new colors-- negative[br]2, negative 2 power-- 0:02:23.820,0:02:30.490 and let me do some other[br]colors-- negative 1, 0, 1. 0:02:30.490,0:02:33.620 I'll do one more, and then 2. 0:02:33.620,0:02:36.660 So once again, this is[br]a little nontraditional, 0:02:36.660,0:02:38.890 where I'm filling in the[br]dependent variable first. 0:02:38.890,0:02:40.300 But the way that[br]we've written it over 0:02:40.300,0:02:42.290 here, it's actually given[br]the dependent variable, 0:02:42.290,0:02:44.750 it's easy to figure out what[br]the independent variable needs 0:02:44.750,0:02:47.130 to be for this[br]logarithmic function. 0:02:47.130,0:02:50.420 So, what x gives me[br]a y of negative 2? 0:02:50.420,0:02:52.640 What x gives me--[br]what does x have 0:02:52.640,0:02:55.430 to be for y to be[br]equal to negative 2? 0:02:55.430,0:02:59.590 Well, 5 to the negative 2 power[br]is going to be equal to x. 0:02:59.590,0:03:04.480 So 5 to the negative[br]2 is 1 over 25. 0:03:04.480,0:03:07.440 So we get 1 over 25. 0:03:07.440,0:03:09.000 If we go back to[br]this earlier one, 0:03:09.000,0:03:13.200 if we say log base[br]5 of 1 over 25, 0:03:13.200,0:03:16.530 what power do I have to[br]raise 5 to to get 1 over 25? 0:03:16.530,0:03:19.420 We'll have to raise it[br]to the negative 2 power. 0:03:19.420,0:03:22.110 Or you could say 5[br]to the negative 2 0:03:22.110,0:03:24.460 is equal to 1 over 25. 0:03:24.460,0:03:27.960 These are saying the[br]exact same thing. 0:03:27.960,0:03:30.160 Now let's do another one. 0:03:30.160,0:03:34.100 What happens when I raise[br]5 to the negative 1 power? 0:03:34.100,0:03:35.487 I get one fifth. 0:03:35.487,0:03:37.320 So if we go to this[br]original one over there, 0:03:37.320,0:03:42.670 we're just saying that[br]log base 5 of one fifth. 0:03:42.670,0:03:44.210 Want to be careful. 0:03:44.210,0:03:46.810 This is saying, what power[br]do I have to raise 5 to 0:03:46.810,0:03:48.130 in order to get one fifth. 0:03:48.130,0:03:50.640 We'll have to raise it[br]to the negative 1 power. 0:03:53.150,0:03:55.080 What happens when I take[br]5 to the 0'th power? 0:03:55.080,0:03:57.249 I get one. 0:03:57.249,0:03:59.290 And so this relationship--[br]This is the same thing 0:03:59.290,0:04:02.650 as saying log base 5 of 1. 0:04:02.650,0:04:05.430 What power do I have[br]to raise 5 to to get 1? 0:04:05.430,0:04:08.910 I just have to raise[br]it to the 0th power. 0:04:08.910,0:04:10.660 Let's do the next two. 0:04:10.660,0:04:13.470 What happens when I raise[br]5 to the first power? 0:04:13.470,0:04:17.180 Well, I get 5 So if you go look[br]over here, that's just saying, 0:04:17.180,0:04:20.410 log, what power do I have[br]to raise 5 to to get 5? 0:04:20.410,0:04:23.880 We'll have to just raise[br]it to the first power. 0:04:23.880,0:04:28.800 And then finally, if I[br]take 5 squared, I get 25. 0:04:28.800,0:04:31.680 So when you look at it from[br]the logarithmic point of view, 0:04:31.680,0:04:34.410 you say, well, what power[br]do I have to raise 5 to 0:04:34.410,0:04:36.020 to get to 25? 0:04:36.020,0:04:38.930 We'll have to raise it[br]to the second power. 0:04:38.930,0:04:41.830 So I took the inverse of[br]the logarithmic function. 0:04:41.830,0:04:43.550 I wrote it as an[br]exponential function. 0:04:43.550,0:04:47.270 I switched the dependent[br]and independent variables, 0:04:47.270,0:04:50.760 so I can derive nice clean x's[br]that will give me nice clean 0:04:50.760,0:04:51.734 y's. 0:04:51.734,0:04:54.150 Now with that out of the way,[br]but I do want to remind you, 0:04:54.150,0:04:57.497 I could have just picked[br]random numbers over here, 0:04:57.497,0:04:59.830 but then I would have probably[br]gotten less clean numbers 0:04:59.830,0:05:00.260 over here. 0:05:00.260,0:05:01.510 I would have had to[br]use a calculator. 0:05:01.510,0:05:03.093 The only reason why[br]I did it this way, 0:05:03.093,0:05:06.810 is so I get nice clean results[br]that I can plot by hand. 0:05:06.810,0:05:08.740 So let's actually graph it. 0:05:08.740,0:05:10.910 Let's actually graph[br]this thing over here. 0:05:10.910,0:05:13.570 So the y's go between[br]negative 2 and 2. 0:05:13.570,0:05:18.650 The x's go from 1/25th[br]all the way to 25. 0:05:18.650,0:05:22.680 So let's graph it. 0:05:22.680,0:05:30.050 So that is my y-axis,[br]and this is my x-axis. 0:05:30.050,0:05:32.116 Draw it like that. 0:05:32.116,0:05:34.190 That is my x-axis. 0:05:34.190,0:05:37.340 And then the y's start at 0. 0:05:37.340,0:05:42.520 Then, you get to[br]positive 1, positive 2. 0:05:42.520,0:05:44.940 And then you have negative 1. 0:05:44.940,0:05:47.230 And you have negative 2. 0:05:47.230,0:05:49.720 And then on the x-axis,[br]it's all positive. 0:05:49.720,0:05:53.180 And I'll let you think about[br]whether the domain here 0:05:53.180,0:05:56.080 is-- well, when you[br]think about it-- 0:05:56.080,0:05:58.270 is a logarithmic[br]function defined 0:05:58.270,0:06:03.050 for an x that is not positive? 0:06:03.050,0:06:07.190 So is there any power that I can[br]raise five to that I can get 0? 0:06:07.190,0:06:08.370 No. 0:06:08.370,0:06:11.120 You could raise five to an[br]infinitely negative power 0:06:11.120,0:06:13.720 to get a very, very, very, very[br]small number that approaches 0:06:13.720,0:06:15.840 zero, but you can[br]never get-- there's 0:06:15.840,0:06:18.260 no power that you can[br]raise 5 to to get 0. 0:06:18.260,0:06:19.799 So x cannot be 0. 0:06:19.799,0:06:21.590 And there's no power[br]then you could raise 5 0:06:21.590,0:06:24.030 to get another negative number. 0:06:24.030,0:06:25.785 So x can also not be[br]a negative number. 0:06:25.785,0:06:28.160 So the domain of this function[br]right over here-- and this 0:06:28.160,0:06:30.410 is relevant, because we want[br]to think about what we're 0:06:30.410,0:06:33.500 graphing-- the domain here is[br]x has to be greater than zero. 0:06:33.500,0:06:35.230 Let me write that down. 0:06:35.230,0:06:39.675 The domain here is that x[br]has to be greater than 0. 0:06:39.675,0:06:41.300 So we're only going[br]to be able to graph 0:06:41.300,0:06:45.660 this function in[br]the positive x-axis. 0:06:45.660,0:06:48.380 So with that out of the[br]way, x gets as large as 25. 0:06:48.380,0:06:51.150 So let me graph-- we[br]put those points here. 0:06:51.150,0:06:57.430 So that is 5, 10,[br]15, 20, and 25. 0:06:57.430,0:06:58.884 And then let's plot these. 0:06:58.884,0:07:00.050 So the first one is in blue. 0:07:00.050,0:07:02.720 When x is 1/25 and[br]y is negative 2-- 0:07:02.720,0:07:06.090 When x is 1/25 so[br]1 is there-- 1/25 0:07:06.090,0:07:09.940 is going to be really close to[br]there-- Then y is negative 2. 0:07:09.940,0:07:12.680 So it's going to be[br]like right over there, 0:07:12.680,0:07:14.190 not quite at the y-axis. 0:07:14.190,0:07:17.040 We're at 1/25 to the[br]right of the y-axis. 0:07:17.040,0:07:17.996 But pretty close. 0:07:17.996,0:07:19.120 So that's right over there. 0:07:19.120,0:07:23.680 That is 1 over 25, comma[br]negative 2 right over there. 0:07:23.680,0:07:26.270 Then, when x is one[br]fifth, which is slightly 0:07:26.270,0:07:30.550 further to the right, one[br]fifth y is negative 1. 0:07:30.550,0:07:32.880 So right over there. 0:07:32.880,0:07:36.950 So this is one[br]fifth, negative 1. 0:07:36.950,0:07:40.290 Then when x is 1, y is 0. 0:07:40.290,0:07:43.310 So 1 might be right there. 0:07:43.310,0:07:46.550 So this is the point 1,0. 0:07:46.550,0:07:50.630 And then when x is 5, y is 1. 0:07:50.630,0:07:53.180 When x is 5, I covered it[br]over here, when this is five, 0:07:53.180,0:07:56.530 y is 1. 0:07:56.530,0:07:59.280 So that's the point 5,1. 0:07:59.280,0:08:02.200 And then finally,[br]when x is 25, y is 2. 0:08:08.060,0:08:11.140 So this is 25,2. 0:08:11.140,0:08:13.050 And then I can[br]graph the function. 0:08:13.050,0:08:17.030 And I'll do it-- let me do it[br]in a color-- I'll use this pink. 0:08:17.030,0:08:23.550 So as x gets super, super,[br]super, super small, y goes 0:08:23.550,0:08:25.820 to negative infinity. 0:08:25.820,0:08:30.490 It gets really small-- to[br]get x's or as x becomes-- 0:08:30.490,0:08:34.179 if you say what power do[br]you have to raise 5 to 0:08:34.179,0:08:36.539 to get 0.0001? 0:08:36.539,0:08:38.530 It has to be very, very,[br]very negative power. 0:08:38.530,0:08:43.090 So y is going to be very[br]negative as we approach 0. 0:08:43.090,0:08:47.390 And then it kind of[br]moves up like that. 0:08:47.390,0:08:53.110 And then starts to kind of[br]curve to the right like that. 0:08:53.110,0:08:54.840 And this thing[br]right over here, is 0:08:54.840,0:08:58.890 going to keep going down at[br]a steeper and steeper rate. 0:08:58.890,0:09:03.160 And it's never going[br]to quite touch. 0:09:03.160,0:09:04.060 the y-axis. 0:09:04.060,0:09:06.370 It's going to get closer[br]and closer to the y-axis. 0:09:06.370,0:09:09.840 But it's never going[br]to be quite touch it.