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# Logarithmic Functions

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### Logarithmic Functions

1. 1. Logarithmic Functions
2. 2. The logarithmic function to the base a, where a > 0 and a ≠ 1 is defined: y = logax if and only if x = a y logarithmic form exponential form When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to. Convert to log form: 16 = 4 Convert to exponential form: 1 log 2 = −3 8 2 log 416 = 2 1 2 = 8 −3
3. 3. LOGS = EXPONENTS With this in mind, we can answer questions about the log: log 2 16 = 4 This is asking for an exponent. What exponent do you put on the base of 2 to get 16? (2 to the what is 16?) 1 log 3 = −2 9 What exponent do you put on the base of 3 to get 1/9? (hint: think negative) log 4 1 = 0 1 1 2 log33 33 = log 2 What exponent do you put on the base of 4 to get 1? When working with logs, re-write any radicals as rational exponents. What exponent do you put on the base of 3 to get 3 to the 1/2? (hint: think rational)
4. 4. Example 1 Solve for x: log 6 x = 2 Solution: Let’s rewrite the problem in exponential form. 6 =x 2 We’re finished !
5. 5. Example 2 1 Solve for y: log 5 =y 25 Solution: Rewrite the problem in exponential form. 1 5 = 25 y 5y = 5− 2 y = −2  1  Since  = 5− 2   25 
6. 6. Example 3 Evaluate log3 27. Solution: Try setting this up like this: log3 27 = y Now rewrite in exponential form. 3y = 27 3y = 33 y=3
7. 7. Example 4 Evaluate: log7 7 2 Solution: log7 7 = y 2 First, we write the problem with a variable. 7y = 72 Now take it out of the logarithmic form y=2 and write it in exponential form.
8. 8. Example 5 Evaluate: 4 log 4 16 Solution: 4 log 4 16 =y First, we write the problem with a variable. log4 y = log4 16 Now take it out of the exponential form and write it in logarithmic form. Just like 2 = 8 converts to log2 8 = 3 3 y = 16
9. 9. Finally, we want to take a look at the Property of Equality for Logarithmic Functions. Suppose b > 0 and b ≠ 1. Then logb x1 = log b x 2 if and only if x1 = x 2 Basically, with logarithmic functions, if the bases match on both sides of the equal sign , then simply set the arguments equal.
10. 10. Example 1 Solve: log3 (4x +10) = log3 (x +1) Solution: Since the bases are both ‘3’ we simply set the arguments equal. 4x +10 = x +1 3x +10 = 1 3x = − 9 x= −3
11. 11. Example 2 Solve: log8 (x −14) = log8 (5x) 2 Solution: Since the bases are both ‘8’ we simply set the arguments equal. 2 x −14 = 5x x 2 − 5x −14 = 0 Factor (x − 7)(x + 2) = 0 (x − 7) = 0 or (x + 2) = 0 x = 7 or x = −2 continued on the next page
12. 12. Example 2 continued Solve: log8 (x −14) = log8 (5x) 2 Solution: x = 7 or x = −2 It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. Let’s end this lesson by taking a closer look at this.
13. 13. Our final concern then is to determine why logarithms like the one below are undefined. log 2 (−8) Can anyone give us an explanation ?
14. 14. log 2 (−8) = undefined WHY? One easy explanation is to simply rewrite this logarithm in exponential form. We’ll then see why a negative value is not permitted. log 2 (−8) = y First, we write the problem with a variable. 2 =−8 y Now take it out of the logarithmic form and write it in exponential form. What power of 2 would gives us -8 ? 1 2 = 8 and 2 = 8 3 −3 Hence expressions of this type are undefined.
15. 15. Characteristics about the Graph of an Exponential Function f ( x ) = a x a > 1 Characteristics about the Graph of a Log Function f ( x ) = log a x where a > 1 1. Domain is all real numbers 1. Range is all real numbers 2. Range is positive real numbers 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x→ -∞ 2. Domain is positive real numbers 3. There are no y intercepts 4. The x intercept is always (1,0) (x’s and y’s trade places) 5. The graph is always increasing 6. The y-axis (where x = 0) is a vertical asymptote
16. 16. Exponential Graph Graphs of inverse functions are reflected about the line y = x Logarithmic Graph
17. 17. Transformation of functions apply to log functions just like they apply to all other functions so let’s try a couple. up 2 f ( x ) = log10 x f ( x ) = 2 + log10 x Reflect about x axis f ( x ) = − log10 x left 1 f ( x ) = log10 ( x + 1)
18. 18. Remember our natural base “e”? We can use that base on a log. exponent do you log e 2.7182828 = 1 Whatto get 2.7182828? put on e ln Since the log with this base occurs in nature frequently, it is called the natural log and is abbreviated ln. ln 2.7182828 = 1 Your calculator knows how to find natural logs. Locate the ln button on your calculator. Notice that it is the same key that has ex above it. The calculator lists functions and inverses using the same key but one of them needing the 2nd (or inv) button.
19. 19. Another commonly used base is base 10. A log to this base is called a common log. Since it is common, if we don't write in the base on a log it is understood to be base 10. log 100 = 2 1 log = −3 1000 What exponent do you put on 10 to get 100? What exponent do you put on 10 to get 1/1000? This common log is used for things like the richter scale for earthquakes and decibles for sound. Your calculator knows how to find common logs. Locate the log button on your calculator. Notice that it is the same key that has 10x above it. Again, the calculator lists functions and inverses using the same key but one of them needing the 2nd (or inv) button.
20. 20. The secret to solving log equations is to re-write the log equation in exponential form and then solve. log 2 ( 2 x + 1) = 3 2 = 2x +1 3 8 = 2x +1 7 = 2x 7 =x 2 Convert this to exponential form check:  7  log 2  2  + 1 = 3  2      log 2 ( 8) = 3 This is true since 23 = 8
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