Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I wanted to do a couple more examples with the change of base theorem. Recall the change of base theorem is log base a of x equals log base b of x over log base b of a. The purpose of the change of base theorem is to change from one base logarithm to another. So here I'm changing from base a to base b and as you'll see in the next example this can be really useful.
Problem 1 says, how would you graph these on your calculator? And the first problem is y equals log base 5 of x. Now my calculator doesn't have a log base 5 button maybe yours does, but many calculators come with only 2 logarithms as buttons on them; common log and natural log.
So, how would you graph these on your calculator? You'd have to change to a log that your calculator has like the common log, so let me do that. I want to change to the log base 10. By the change of base formula. This would be the log base 10 of x, x in this case is just x over the log base 10 of 5. And if you like you can approximate this. This is the same as 1 over the log base 10 of 5 times the common log of x. 1 over log 5, so that's approximately 1.431. So that's how you graph log base 5 of x on your calculator.
Now the next one log base x of 5, this is not actually a logarithmic function, I know it looks like one. But the fact that the x is in the base rather than in the argument makes it not a log function and you'll see why in a second. I'm going to use the change of base theorem to rewrite it in terms of log base 10. So it will be log base 10 of 5 over log base 10 of x.
So this is the common log of 5, over the common log of x. We're dividing by log of x so this is actually the reciprocal of a logarithmic function. Anyway if I were graphing y equals log base x of 5, this is what I would take into my calculator.
Now here is another problem, simplify log base 2 of 3 times log base 3 of 5 times log base 5 of 8. This looks really tricky, but if you use the change of base theorem it's pretty easy. Now what I can do is, I can change to any base I want, but I'm going to change all of these to the common base of 10. So I'm going to write the log base 2 of 3 as the log of 3 over the log of 2 and the log base 3 of 5, that's the log of 5 times the log of 3. You'll see what's going to happen. I'm going to get a lot of cancellation. And then log base 5 of 8 is log of 8 over log of 5. So the log 3's cancel, the log 5's cancel and I'm left with the log of 8 over the log of 2.
Now if I use the change of base theorem in reverse, this is the same as the log base 2 of 8, 8 is 2 to the third power, the log base 2 of 2 to the third power is 3, so this whole thing is just three, that's the change of base theorem.