Hey guys, so here it's going to become important that you guys remember the different operations that occur when we're dealing with logarithmic functions and natural logarithmic functions. So for example, let's take a look at multiplication. So here we have a log of a * b. Now that log is getting distributed to my A and my B, and because they're multiplying with each other, what that translates to is log of A plus log of B. And the same thing can be seen with our natural log. So ln(AB) gets distributed to both. Because they're multiplying, it really means ln(A)+ln(B).
Next, we're dealing with division. So when we're dividing them, what does that really mean? Well, that means here that log of A / B becomes log of A minus log of B. So when you're multiplying, it really means you're adding each one. But when you're dividing and subtracting, Same thing with my ln(A)-ln(B). What happens when we raise it to a power? Woah, when we raise it to a power. What you need to realize here is that this power can move up front of my log function. So this becomes X times log of A. And same thing for ln(A), X comes up front, so it becomes X (ln(A)).
So there's a little bit of trick functions when it comes to these types of situations. This is going to become important when we're dealing with chapters dealing with chemical kinetics because with those chapters we do have to manipulate problems that contain ln. And then when it comes to pH and POH of acids and bases, that's when we have to take into account log functions. And some of the questions require these types of manipulations, then if we're taking it to the NTH route. So here we have a log of X square root sign and and a inside. So remember that could be two. That could be three. That could be 4. So second root, which is square root, third root, 4th root.
So remember, when we're doing a NTH root, what does that really mean? First of all, what that means? It's log of A to the 1 / X. OK, so when we change that into a, power becomes the reciprocal power, so it becomes 1 / X. And again like we said before, the exponent can come up front, so this equals 1 / X times log of a. And then here, same thing with ln(A), it becomes ln(A)^(1/X). Again, this can come out front, so this equals 1 / X times ln(A). So these are the different manipulations that you need to be aware of that will happen when we get to again, chemical kinetics and when we're dealing with log functions of acids and bases.
Based on what we've seen here, I say solve the following without the use of a calculator. If log of three is equal to approximately 48 and log of two is equal to approximately 30, what will be the value of log of 12? All right, so we have to think of which of these operations we can use. So we want to find log of 12. Now an easy way to to approach this is we can think of what could I do with three and two to give me a value of 12. That's why I gave you those numbers. And the answer is if I do 3 * 2 that gives me 6 and then after I multiply that by two again that gives me 12.
So we can say that log of 12 can be seen as log of 3 * 2 * 2 and we just saw up above. When you have things multiplying together, the law gets distributed to all of them. And because there's multiplication under on being done, that really means it's log of three plus log of two plus log of two. And so all we're going to do now is tell us what these values are. This is 0.48, this is 0.30, and this is .30 O. When we plug all that in, that gives me 1.08 as my final answer, O. That would be my log of 12. And we did it without the use of a calculator because we understand the different relationships that arise when we're dealing with log functions or natural log functions.