Logarithmic Functions The logarithmic function to the base
Logarithmic Functions
The logarithmic function to the base b, where b > 0, x > 0 and b 1 is defined: y = logbx if and only if x = b 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)
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 b>1 Characteristics about the Graph of a Log Function where b > 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 no x value exists that will make the function equal zero (no value added/subtracted outside the function). 4. The y intercept is always (0, 1) because b 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.
Exponential Graphs of inverse functions are reflected about the line y = x Logarithmic Graph
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
Properties of Logarithms Use to solve logarithmic equations by combining logs into one so either the logs can eliminate OR be re-written in exponential form.
Product Property AND Example:
Quotient Property AND Example:
Power Property Example: Equality Property Ex.
Miscellaneous Properties Ex. Ex.
- Slides: 12