College Algebra: Exponential to Log Functions

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So to get us sort of into the spirit of going from exponents to logs and from logs to exponents, let’s just take a look at some very specific number type examples, just to go back and forth a little bit, so nothing profound here. This is a non-profound lecture. Sort of boring. So 25 = … Oh, in fact, you know what? I have an idea. Let’s actually have all these things be interactive, and that way you get a chance to sort of, you know, be boring with me, basically. So here’s what I want you to do. For each fact--25 = 32. You can check that--2 times 2 times 2 five times is in fact 32. That’s fine. What I want you to do is rewrite this exact statement using logs without using exponents. Just remember, the log is an exponent. You have to figure out the base and figure out what the log of the function is. So see if you can convert this into a statement with logs. I’ll do it after you try it. Okay, give it a shot right now.
Okay, well, let’s see how you did. Here’s what I remember. A log is an exponent, so that exponent must equal the log. So I must have that 5 is the exponent. So 5 is log. Right? Log is the exponent that I have to raise the base 2 to, so this must be a base 2, in order to get 32. So these two things are identical. Let’s check. A log is the exponent, the exponent I have to raise 2 to in order to get 32. So 25 = 32. That checks.
Okay, the first one might be sort of tricky, but now after that hopefully you should be able to make some progress on these. 2/3-2. What does that equal? Let’s just do that in our heads. The negative sign will flip this so I get 3/2, but then I square it so I get 9/4. So this equals 9/4. There’s a statement that is true. Now, what I want from you is the analogous statement in terms of logs. Give it a shot right now.
Okay, well, a log is an exponent, and the exponent here is -2, so -2 = the log = the exponent that I have to raise this base to 2/3 in order to get 9/4. So the answer is -2 = log2/3 9/4. That’s how you read that, by the way. You read this as -2 equals log base 2/3 of 9/4. That’s how you’d read this. This you’d say 5 = log base 2 of 32. That’s how you read it. Okay, well let’s go the other way now. Suppose that I give you the log and I want you to convert it back to a fact about exponents. How about this? Log6 6 = 1. See if you can convert that to a statement about exponents where it should be very, very clear that, in fact, this statement is correct. Give it a shot right now.
Okay, well, let’s see. A log is the exponent, so 1 is the exponent that I have to raise the base 6 to in order to get 6. So this is identical to 61 = 6, and that’s a true statement. So this statement is identical to this, because a log is the exponent, so there’s the exponent I have to raise the base to in order to get 6. One last one just for fun. How about this? Here we go. Log 9 = 4. Now, that looks really weird. In fact, maybe that’s not even true, but I want you to convert this into the equivalent exponential fact and see if, in fact, this is really true or not. Maybe it’s not even a true statement. Give it a shot right now. Be careful and work through it.
Okay, well, if you remember my little chant, we should be sort of home free, because a log is an exponent. So 4 is the exponent I have to raise this base to in order to get 9. So this is identical to saying that the 4 should equal 9. Is that true? Well, let’s think about it. The 4 is the same thing as the ( 2)2 because to the fourth is just squared, squared. Let me write that down so you can actually see that. So 4 is just the ( 2)2, because what do you do when you square a square? You multiply and you get 4. Well, the 2 is just 3, so this equals 32, which equals 9, so in fact, this is really a correct statement, and we’re able to see that correct statement just simply by converting this log expression back to an exponent expression. So you can go from exponents to logs and you can go from logs to exponents, just by remembering my little mantra, a log is an exponent.
Okay, hopefully that gives you a better sense of how the logs work and how they sort of interact with the equivalent counterparts of exponents, and now we’ll start to move forward with logs.