1. Unit 11, Part 2: Day 1: Introduction To Logarithms

2. Logarithms were originally developed to simplify complex arithmetic calculations. They were designed to transform multiplicative processes into additive ones. (We use them now to help us solve exponential equations, when the “x” is a part of the exponent.) Logarithms were originally developed to simplify complex arithmetic calculations. They were designed to transform multiplicative processes into additive ones. (We use them now to help us solve exponential equations, when the “x” is a part of the exponent.)

3. If at first this seems like no big deal, then try multiplying 2,234,459,912 and 3,456,234,459. Without a calculator ! Using logarithms, the first number in log form(base 10) has a log value of about 9.349. The second has a log value of about 9.539. ADDING the two values together gets us a log value of 18.888 Converting it back into a normal form (using a table of log values), we get about 7,726,805,851,000,000,000. Clearly, it was a lot easier to add versions of these two numbers and then converting it. If at first this seems like no big deal, then try multiplying 2,234,459,912 and 3,456,234,459. Without a calculator ! Using logarithms, the first number in log form(base 10) has a log value of about 9.349. The second has a log value of about 9.539. ADDING the two values together gets us a log value of 18.888 Converting it back into a normal form (using a table of log values), we get about 7,726,805,851,000,000,000. Clearly, it was a lot easier to add versions of these two numbers and then converting it.

4. Today of course we have calculators and scientific notation to deal with such large numbers. So at first glance, it would seem that logarithms have become obsolete. Today of course we have calculators and scientific notation to deal with such large numbers. So at first glance, it would seem that logarithms have become obsolete.

5. Indeed, they would be obsolete except for one very important property of logarithms. It is called The Power Property and we will learn about it in another lesson. For now we need only to observe that it is an extremely important part of solving exponential equations. Indeed, they would be obsolete except for one very important property of logarithms. It is called The Power Property and we will learn about it in another lesson. For now we need only to observe that it is an extremely important part of solving exponential equations.

6. Our first job is to try to make some sense of logarithms.

7. Our first question then must be: What is a logarithm ?

8. Of course logarithms have a precise mathematical definition just like all terms in mathematics. So let’s start with that.

9. Definition of Logarithm Suppose b>0 and b≠1, there is a number ‘p’ such that: Suppose b>0 and b≠1, there is a number ‘p’ such that: Also, a logarithmic function is the INVERSE of an exponential function.

10. The first, and perhaps the most important step, in understanding logarithms is to realize that they always relate back to exponential equations.

11. You must be able to convert an exponential equation into logarithmic form and vice versa. So let’s get a lot of practice with this ! This is how we convert from one form to another… LOG FORM ↔ EXPONENTIAL FORM log base (value) = exponent ↔ base exponent = value

12. The Log and Loop Trick…

13. Example 1: Solution: We read this as: ”the log base 2 of 8 is equal to 3”.

14. Example 1a: Solution: Read as: “the log base 4 of 16 is equal to 2”.

15. Example 1b: Solution: NOTE: Logarithms can be equal to negative numbers BUT… The base of a logarithm cannot be negative… NOR can we take the log of a negative number!!

16. Okay, so now it’s time for you to try some on your own.

17. Solution:

20. It is also very important to be able to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did. It is also very important to be able to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did.

21. Example 1: Solution:

22. Example 2: Solution:

23. Okay, now you try these next three.

24. Solution: