1. Sections 5.3-5.5

2. Logarithmic Functions (5.3) What is a logarithm??? LOGS ARE POWERS!!!! A logarithm or “ log ” of a number of a certain base is the exponent to which the base of the log must be raised in order to produce the number. The base cannot equal 1 and must be greater than 0. For instance, if log b (x) = c and b≠1 and b >0, then c is the specific exponent to which you must raise b in order to get x : b c = x

3. Logarithmic Functions Why do we need logs? Let’s explore… 3 2 = 9 and 3 3 = 27 but what would we need to raise 3 to in order to get 20?? 3 a = 20 that’s what logs tell us!! a = log 3 20 Which two integers is log 3 20 between? 2 and 3

4. Logarithmic Functions From the definition, we have stated that if log b (x) = c, then b c = x under the conditions that b≠1 and b >0. Why do we need to place any restrictions on b or x so this can make sense? Let’s try some values…

5. log b (x) = c, so b c = x log 2 (8) = c so 2 c = 8 c = 3, so far we are ok log 1 (5) = c so 1 c = 5 Does not exist; 1 c always equals 1 log -2 (8) = c so (-2) c = 8 Does not exist; if c = 3, then (-2) 3 = - 8 log 3 (-9) = c so 3 c = -9 Does not exist; 3 c cannot be negative log 2 (0) = c so 2 c = 0 Does not exist; 2 c cannot equal 0 Summary: b≠1, b>0 and x>0

6. Log Properties (1) log b b = 1 (2) log b 1 = 0 common log has base 10: log(x) = log 10 (x) natural log has base e : ln(x) = log e (x) Therefore… log10 = 1 ln e = 1

7. Practice Evaluate, if possible. If not, state so.

8. y = log b (x - h) + k When graphing logs we first need to identify and graph the asymptote. Earlier we discovered that the argument inside the log must be greater than 0. Therefore, x > h so the domain is (h, +∞) and there must be an asymptote at x = h The range is all real numbers Now find three points; the simplest values are when x - h = 1 and when x - h = b

9. Graph of a Logarithmic Function Graph of y = log b (x) when b>1 Can you state any characteristics? –Asymptotes, x - intercepts, Domain, Range Graph of y = log b (x) when 0<b<1

10. Practice Graph. State the domain and range.

11. Change of Base Formula This formula allows us to compute logs using the calculator, by converting to base 10 or e. Example: