Download presentation

Presentation is loading. Please wait.

Published byTeresa Welch Modified over 4 years ago

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:

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google