Logarithmic Functions

Convert to exponential form: 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: Convert to exponential form:

LOGS = EXPONENTS With this in mind, we can answer questions about the log: 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?) What exponent do you put on the base of 3 to get 1/9? (hint: think negative) 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)

In the last section we learned about the graphs of exponentials. Logs and exponentials are inverse functions of each other so let’s see what we can tell about the graphs of logs based on 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 Graph of an Exponential Function a > 1 Characteristics about the Graph of a Log Function where a > 1 1. Domain is all real numbers 1. Range is all real numbers 2. Range is positive real numbers 2. Domain 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 3. There are no y intercepts 4. The x intercept is always (1,0) (x’s and y’s trade places) 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing 5. The graph is always increasing 6. The x-axis (where y = 0) is a horizontal asymptote for x  -  6. The y-axis (where x = 0) is a vertical asymptote

Logarithmic Graph Exponential Graph Graphs of inverse functions are reflected about the 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 Reflect about x axis left 1

ln Remember our natural base “e”? We can use that base on a log. What exponent do you put on e to get 2.7182828? ln Since the log with this base occurs 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 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. 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.

The secret to solving log equations is to re-write the log equation in exponential form and then solve. Convert this to exponential form check: This is true since 23 = 8