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 formWhen you convert an exponential to log form, notice that theexponent in the exponential becomes what the log is equal to.Convert to log form: 16 4 2 log 416 2Convert to exponential form: 31 1 log 2 3 2 8 8
LOGS = EXPONENTSWith this in mind, we can answer questions about the log: This is asking for an exponent. Whatlog 2 16 4 exponent do you put on the base of 2 to get 16? (2 to the what is 16?) 1 What exponent do you put on the base oflog 3 2 3 to get 1/9? (hint: think negative) 9 log 4 1 0 What exponent do you put on the base of 4 to get 1? 1 When working with logs, re-write any 1 log33 33 log 2 radicals as rational exponents. What exponent do you put on the base of 2 3 to get 3 to the 1/2? (hint: think rational)
Logs and exponentials are inverse functions of each otherso let’s see what we can tell about the graphs of logs basedon what we learned about the graphs of exponentials. Recall that for functions and their inverses, x’s and y’s trade places. So anything that was true about x’s or the domain of a function, will be true about y’s or the range of the inverse function and vice versa. Let’s look at the characteristics of the graphs of exponentials then and see what this tells us about the graphs of their inverse functions which are logarithms.
Characteristics about the Characteristics about theGraph of an Exponential Graph of a Log FunctionFunction f x a x a > 1 f x log a x where a > 11. Domain is all real numbers 1. Range is all real numbers2. Range is positive real 2. Domain is positive realnumbers numbers3. There are no x intercepts 3. There are no y interceptsbecause there is no x valuethat you can put in thefunction to make it = 04. The y intercept is always 4. The x intercept is always(0,1) because a 0 = 1 (1,0) (x’s and y’s trade places)5. The graph is always 5. The graph is alwaysincreasing increasing6. The x-axis (where y = 0) is 6. The y-axis (where x = 0) isa horizontal asymptote for a vertical asymptote x-
Exponential Graph Logarithmic GraphGraphs ofinversefunctions arereflected aboutthe line y = x
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 log 10 xf x 2 log 10 x Reflect about x axis f x log 10 x left 1f x log10 x 1
Remember our natural base “e”? We can use that base on a log.log e 2.7182828 1 Whatto get 2.7182828? put on e exponent do you ln Since the log with this base occursln 2.7182828 1 in nature frequently, it is called the natural log and is abbreviated ln. 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.
Another commonly used base is base 10. A log to this base is called a common log. Since it is common, if we dont write in the base on a log it is understood to be base 10. log 100 2 What exponent do you put on 10 to get 100? 1 log 3 What exponent do you put on 10 to get 1/1000? 1000This common log is used for things like the richterscale for earthquakes and decibles for sound.Your calculator knows how to find common logs.Locate the log button on your calculator. Notice that itis the same key that has 10x above it. Again, thecalculator lists functions and inverses using the samekey but one of them needing the 2nd (or inv) button.
The secret to solving log equations is to re-write thelog equation in exponential form and then solve. log 2 2 x 1 3 Convert this to exponential form check: 2 2x 1 3 7 log 2 2 1 3 2 8 2x 1 7 2x log 2 8 3 7 x This is true since 23 = 8 2
AcknowledgementI wish to thank Shawna Haider from Salt Lake Community College, UtahUSA for her hard work in creating this PowerPoint.www.slcc.eduShawna has kindly given permission for this resource to be downloadedfrom www.mathxtc.com and for it to be modified to suit the WesternAustralian Mathematics Curriculum.Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au