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Dialogue: 0,0:00:00.14,0:00:03.06,Default,,0000,0000,0000,,INSTRUCTOR: In this video, we're briefly going to go over
Dialogue: 0,0:00:03.06,0:00:06.80,Default,,0000,0000,0000,,some properties of logs that you need to be familiar with.
Dialogue: 0,0:00:06.80,0:00:09.20,Default,,0000,0000,0000,,The first one is the power rule.
Dialogue: 0,0:00:09.20,0:00:12.15,Default,,0000,0000,0000,,Let's say if you have log A raised to the N.
Dialogue: 0,0:00:12.15,0:00:14.67,Default,,0000,0000,0000,,You can move the exponent in front.
Dialogue: 0,0:00:14.67,0:00:18.10,Default,,0000,0000,0000,,This is equal to N log A.
Dialogue: 0,0:00:19.31,0:00:22.98,Default,,0000,0000,0000,,The second is the product rule.
Dialogue: 0,0:00:22.98,0:00:30.58,Default,,0000,0000,0000,,Log A times log B is equal to log A plus log B.
Dialogue: 0,0:00:30.83,0:00:33.54,Default,,0000,0000,0000,,Then you have the quotient rule.
Dialogue: 0,0:00:33.54,0:00:43.68,Default,,0000,0000,0000,,Log A divided by log B is log A minus log B.
Dialogue: 0,0:00:43.68,0:00:47.80,Default,,0000,0000,0000,,We can use this in order to evaluate logs.
Dialogue: 0,0:00:47.80,0:00:56.79,Default,,0000,0000,0000,,For instance, let's say if we want to simplify log base 5 that's 5 raised to 7.
Dialogue: 0,0:00:56.79,0:01:00.88,Default,,0000,0000,0000,,What we can do is move the 7 to the front.
Dialogue: 0,0:01:00.88,0:01:06.80,Default,,0000,0000,0000,,This is equal to 7 log base 5 of 5.
Dialogue: 0,0:01:06.80,0:01:09.68,Default,,0000,0000,0000,,A log base 5 of 5 because they're the same,
Dialogue: 0,0:01:09.68,0:01:13.73,Default,,0000,0000,0000,,it's equal to 1 and 7 times 1 is 7.
Dialogue: 0,0:01:13.84,0:01:16.76,Default,,0000,0000,0000,,Let's try another example like that.
Dialogue: 0,0:01:16.76,0:01:19.65,Default,,0000,0000,0000,,Log base 2 of 8 raised to fifth power.
Dialogue: 0,0:01:21.29,0:01:24.90,Default,,0000,0000,0000,,Let's move the 5 to the front.
Dialogue: 0,0:01:24.90,0:01:30.88,Default,,0000,0000,0000,,This is equal to 5 times log base 2 of 8.
Dialogue: 0,0:01:30.88,0:01:34.54,Default,,0000,0000,0000,,Now, how many 2s do we need to multiply to get to 8?
Dialogue: 0,0:01:34.54,0:01:39.12,Default,,0000,0000,0000,,We need to multiply 3 2s, 2 to the third is 8,
Dialogue: 0,0:01:39.12,0:01:41.25,Default,,0000,0000,0000,,so log base 2 of 8 is 3,
Dialogue: 0,0:01:41.25,0:01:45.39,Default,,0000,0000,0000,,and 5 times 3 is 15.
Dialogue: 0,0:01:45.39,0:01:48.48,Default,,0000,0000,0000,,Now let's work on this example.
Dialogue: 0,0:01:48.48,0:01:51.28,Default,,0000,0000,0000,,Log base 2 of 16 times 8.
Dialogue: 0,0:01:52.10,0:01:56.42,Default,,0000,0000,0000,,We can use the product rule to separate the 16 and the 8.
Dialogue: 0,0:01:56.42,0:02:05.66,Default,,0000,0000,0000,,If you recall, log of A times B equals log A plus log B.
Dialogue: 0,0:02:05.66,0:02:11.44,Default,,0000,0000,0000,,Therefore, log base 2 of 16 times 8 is going to be log base 2
Dialogue: 0,0:02:11.44,0:02:17.49,Default,,0000,0000,0000,,of 16 plus log base 2 of 8.
Dialogue: 0,0:02:17.49,0:02:19.14,Default,,0000,0000,0000,,Now, log base 2 of 16,
Dialogue: 0,0:02:19.14,0:02:21.88,Default,,0000,0000,0000,,that's 4 because 2 to the fourth power is 16.
Dialogue: 0,0:02:21.88,0:02:24.15,Default,,0000,0000,0000,,Log base 2 of 8 is 3.
Dialogue: 0,0:02:24.15,0:02:27.97,Default,,0000,0000,0000,,The final answer is 4 plus 3 which is 7.
Dialogue: 0,0:02:30.71,0:02:37.57,Default,,0000,0000,0000,,Try this one. Log base 3 of 27 times 81.
Dialogue: 0,0:02:37.97,0:02:41.51,Default,,0000,0000,0000,,First, let's separate it into two logs.
Dialogue: 0,0:02:41.51,0:02:47.18,Default,,0000,0000,0000,,This is log base 3 of 27 plus log base 3 of 81.
Dialogue: 0,0:02:47.18,0:02:51.72,Default,,0000,0000,0000,,3 to the third power is 27, so log base 3 of 27 is 3.
Dialogue: 0,0:02:51.72,0:02:55.90,Default,,0000,0000,0000,,Log base 3 of 81 is 4 because 3 to the fourth power is 81,
Dialogue: 0,0:02:55.90,0:03:01.30,Default,,0000,0000,0000,,and this adds to 7. What about division?
Dialogue: 0,0:03:01.30,0:03:07.39,Default,,0000,0000,0000,,What is log base 4 of 256 minus 64?
Dialogue: 0,0:03:08.18,0:03:10.94,Default,,0000,0000,0000,,Using the quotient rule,
Dialogue: 0,0:03:10.94,0:03:17.97,Default,,0000,0000,0000,,log A divided by B is equal to log A minus log B.
Dialogue: 0,0:03:17.97,0:03:20.85,Default,,0000,0000,0000,,Therefore, this is going to be
Dialogue: 0,0:03:20.85,0:03:31.44,Default,,0000,0000,0000,,log base 4 of 256 minus log base 4 of 64.
Dialogue: 0,0:03:31.44,0:03:33.04,Default,,0000,0000,0000,,4 to the fourth power is 256,
Dialogue: 0,0:03:33.04,0:03:35.56,Default,,0000,0000,0000,,and 4 to the third power is 64,
Dialogue: 0,0:03:35.56,0:03:40.54,Default,,0000,0000,0000,,and 4 minus 3 is 1. That's going to be the answer.
Dialogue: 0,0:03:41.84,0:03:47.54,Default,,0000,0000,0000,,Try this one, log base 2 of 128 over 8.
Dialogue: 0,0:03:47.54,0:03:54.99,Default,,0000,0000,0000,,This is equal to log base 2 of 128 minus log base 2 of 8.
Dialogue: 0,0:03:54.99,0:04:01.96,Default,,0000,0000,0000,,Now, 2 to the seventh power is 128 and 2 to the third is 8.
Dialogue: 0,0:04:01.96,0:04:06.43,Default,,0000,0000,0000,,Seven minus 3 is 4 and that's going to be the answer.
Dialogue: 0,0:04:08.33,0:04:17.50,Default,,0000,0000,0000,,Here's the last example. Log base 2 of 128 times 64 divided by 8 times 16 raised to the fifth power.
Dialogue: 0,0:04:22.01,0:04:24.42,Default,,0000,0000,0000,,The first thing we should do is move the
Dialogue: 0,0:04:24.42,0:04:30.78,Default,,0000,0000,0000,,five and it's going to be distributed to everything.
Dialogue: 0,0:04:30.78,0:04:34.87,Default,,0000,0000,0000,,Now, every number that's on top will have a positive sign.
Dialogue: 0,0:04:34.87,0:04:37.73,Default,,0000,0000,0000,,The numbers on the bottom will have a negative sign.
Dialogue: 0,0:04:37.73,0:04:46.82,Default,,0000,0000,0000,,It's going to be log base 2 of 128 plus log base 2 of 64 since they're multiplied to each other,
Dialogue: 0,0:04:46.82,0:04:50.24,Default,,0000,0000,0000,,minus log base 2 of 8,
Dialogue: 0,0:04:50.24,0:04:52.22,Default,,0000,0000,0000,,since it's divided by,
Dialogue: 0,0:04:52.22,0:04:56.03,Default,,0000,0000,0000,,and also minus log base 2 of 16.
Dialogue: 0,0:04:56.25,0:04:59.93,Default,,0000,0000,0000,,Log base 2 of 128, we know it's 7,
Dialogue: 0,0:04:59.93,0:05:03.08,Default,,0000,0000,0000,,2 to the sixth power is 64,
Dialogue: 0,0:05:03.08,0:05:09.58,Default,,0000,0000,0000,,2 to the third is 8, 2 to the fourth is 16.
Dialogue: 0,0:05:09.58,0:05:12.30,Default,,0000,0000,0000,,Negative 3 and negative 4 adds up to negative 7,
Dialogue: 0,0:05:12.30,0:05:14.90,Default,,0000,0000,0000,,which cancels with the positive 7.
Dialogue: 0,0:05:14.90,0:05:19.31,Default,,0000,0000,0000,,The final answer is 5 times 6 which is 30.