### Stony Brook MAT 123 Fall 2015 Rules for manipulating logarithms

 Start Now we're going to learn about how to manipulate logs Using the log laws. Just the way we have rules of exponents, we have rules of logarithms. So..when you have x to the a ..and you multiple it by x to the b ..you get x to the a plus b. Remember logarithms are powers. So if you have the log of two things multiplied together, that's like when you have x to the a times x to the b. 0:36 That becomes the log of a plus the log of b Because you are gonna add the powers, and logarithms are powers. What if we have x to the a divided by x to the b? Which is x to the a minus b. Now were going to have log of a over b. 1:01 Which is log of a minus log of b And the third rule, when we have x to the a raised to the b We multiply the powers So if log of a to the b, that is the same as multiplying b times the log of a So those are our three laws. Now let's do a couple of examples.. 1:43 If I have the log of 4 times 5 Which is the log of 20. That's going to be the log of 4 times 5, is log of 4 plus the log of 5. 2:05 And if we take out our calculator and you find the log of 20 and then you take the log of 4 and the log of 5 and add them together you will get the log of 20. So lets do it. We take out the calculator ..and we have the log of 4 ...is about .6021 and the log of 5 is about .6990. So if we add those together... 2:42 We get 1.3011. By the way these are logs base 10. Now whats the log of 20... We go to the calculator, and you get 1.3010 but that's for rounding So you can see that if you take a look at the log laws.. 3:01 The log of two things multiplied together is the first log plus the second one. Now another thing we can do is use these log laws to break up something complicated. So suppose we have the log of x to the fifth times y to the third Well that's the same as saying log of x to the fifth plus log of y to the third and that's using our rule, that if two things are multiplied together, it's the log of the first plus the log of the second. 3:37 But we also have our rule that says a log of a number raised to a power is multiplied by the power times the log So log of x to the fifth times y cubed is the same as 5 times the log of x plus 3 times the log of y. 4:00 Suppose I had the log of x to the seventh over y to the sixth.. Well now I can break that into log of x to the seventh minus log of y to the sixth because when I am dividing, I subtract the logs. And now, if I put the powers infront And I get this.... 4:30 So here's a nice example.. If the log of a is 3, and the log of b is 7 What is the log of a cubed, b squared? 5:06 Well the log of a cubed times b squared.. Is the log of a cubed plus the log of b squared Now I could take the 3 and put it in front.. and get 3 times the log of a plus 2 times the log of b. 5:33 and now i know, log of a is 3 and log of b is 7. So this gives you 9 plus 14 ..is 23 So notice..we can take our log laws and break up something complicated and make it into something much simpler.