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# Properties of Logarithms - Problem 3

###### Norm Prokup

###### Norm Prokup

**Cornell University**

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.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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