Transcription
In this lesson, we're going to focus on evaluating logs. So, what is log base 2 of 16? What is that equal to? 2 raised to the what power is 16? How many twos do you have to multiply to get to 16? 2 * 2 * 2 * 2 = 16. So, therefore, 2^4 = 16. So, log base 2 of 16 is equal to 4.
So now that you know that, what is log base 3 of 27? How many threes do you need to multiply to get to 27? You need to multiply 3 * 3 * 3 = 27. So log base 3 of 27 is three.
Now, what about log base 5 of 25? How many fives do you have to multiply to get to 25? You need to multiply two fives. 5^2 = 25.
Now, what about log base 4 of 1? Log of one is always equal to zero.
What about log base 7 of 7? 7 to the first power is 7. So, if these numbers are the same, they will cancel; it's simply one.
Now, what is log of 1,000? What should you do if you're not given a base? If there is no base, it's always assumed to be a 10. So, how many tens do you have to multiply together to get to 1,000? You need to multiply 10 three times. This will give you three zeros, and that's equal to 1,000. So, 10^3 = 1,000. Therefore, log base 10 of 1,000 is three.
Now, what about log base 10 of 100? 10 * 10 = 100. So, you need to multiply two tens to get to 100. So, it's two.
Now, what about log of 0.1? What is that equal to? So, the base is 10. It turns out that 10^-1 = 0.1. So, this is equal to -1.
So here are some things to know: log of one is always zero; log of 10, when the base is 10, this is equal to 1. Notice the pattern: log of 100 is equal to 2, and log of 1,000 is equal to 3. Notice that when you have two zeros, it's two; log 1,000 has three zeros, it's three.
So, based on that, what is log of 1 million? Notice that there are six zeros; this is going to equal six. Log of 10,000 has four zeros, so this is equal to four.
Now, what about log of 0.1? This is -1. Log of 0.01 is going to be -2. Log of 0.001 that's going to equal -3. And log of 0.0001 is equal to -4, assuming, of course, the base is 10, which it is if no base is written. So, keep that in mind.
Now, let's work on some other examples. Let's say if you were to see something that looks like this on the homework, what do you think the answer is? What is 7 log base 7 of 38 equal to? Whenever you see this, simply cancel the seven, and it's going to equal to whatever you see here, which is 38.
So, knowing that, try these two: What is 5 log base 5 of 14? And also 8 log base 8 of y? So, this is going to equal 14, and this one is simply going to equal y.
Now, let's try some more examples. What is log base 3 of 9 equal to? Well, we know we have to multiply two threes to get to 9. 3^2 = 9. So, the answer is two.
Now, what is log base 3 of 1/9? How will that change the answer? It turns out that this is going to be equal to -2. 3^2 = 9; 3^-2 = 1/9. The negative exponent will cause the nine to move from the top to the bottom.
Now, what if you reverse the numbers? If log base 3 of 9 is 2, what's log base 9 of 3? So, here's a hint: there's going to be a two involved. It turns out that it's 1/2. You need to flip the fraction if you reverse 3 and 9.
Finally, what is log base 9 of 1/3? This is going to be -1/2. So, anytime you have a fraction, notice that it's going to be negative. Anytime this number is larger, notice that you'll have a fraction. Now, if this number is larger than this one, not including the fraction, if you just compare 9 and 3, then the answer typically will be a number that's greater than one; it's not going to be a fraction.
Let's work on some more examples. Try these: log base 2 of 32, log base 2 of 1/32, log base 32 of 2, and log base 32 of 1/2.
Now, what is log base 2 of 32? How many twos do you have to multiply to get to 32? It takes five twos to get to 32. So, this is five.
Now, if 2^5 = 32, that means 2^-5 = 1/32. So, this is going to be -5.
Now, 32 raised to the what power is two? It turns out that the fifth root of 32 is two. So, this is going to be 1/5.
Now, if 32^(1/5) = 2, 32^(-1/5) = 1/2. And so that's this answer. So, there's going to be a five somewhere; it's either positive five or negative five, positive 1/5 or negative 1/5. And if you convert it to its exponential form, it can help you to determine which answer is correct.
So, let me give you some practice problems, but mix in the order. Go ahead and find the answers to these questions: 2 to the what power is 8? 2^3 = 8. 2 * 2 * 2 = 8. Now we know that 3^2 = 9, but 3^-2 = 1/9. So that's -2. The square root of 25 is 5, so 25^(1/2) = 5. The square root of 64 is 8, so 64^(1/2) = 8. Now we know 2^4 = 16, so the fourth root of 16 is 2. So 16^(1/4) = 2. Therefore, 16^(-1/4) = 1/2. Now 3^4 = 81, so the fourth root of 81 is 3. So if 81^(1/4) = 3, 81^(-1/4) = 1/3. And so that's this for.