Section 3. Logarithmic Functions and Their Graphs Look at the graph of f(x) = x Does this graph pass the Horizontal Line Test? es What does this mean? that its inverse is a function Find the inverse of = a x. (switch x and and solve for ) = a x x = a *We don t know how to solve for!!!
Section 3. Definition: The function given b f(x) = log a x, where x > 0, a > 0, and a, is called the logarithmic function with base a. It is the inverse of the exponential function f(x) = a x. *Thus, = log a x is equivalent x = a. (The both name the inverse of = a x.) Working definition: The log of a number is the exponent that ou put on the base to get that number. (Memorize this!) **Remember: The logarithm is an exponent. Examples: Solve. (a) log 3 (b) log 3 3 3 5 0 (c) log 0 00 0 00 0 0
Section 3. (d) log 4 (e) Evaluate f x) log x ( for x=8 4 ( ) f (8) 8 3 log log 8 8 (f) Evaluate log 0.5 (g) Evaluate log 3 8 0.5 4 3 3 8 3 4 4 **To find the log of a number, write it in exponential form, get the bases the same, and the set the exponents equal and solve. 3
Section 3. Common Logarithms Because we are working in a base 0 number sstem, we call the logarithmic function with base 0 the common logarithmic function. This is the function that corresponds to the LOG button on our calculators. The common logarithmic function is one function for which we need not write the base. Examples: Find the following: (a) log0 (b) log (c) log3. 5 (d) log( ) 4-0.60 0.544 Error NONREAL ANS Note: 0 = - will never happen. Graphing Logarithmic Functions On our calculator, graph = 0 x = log x = x 4
Section 3. = 0 x = x = log x *Notice that for = log x, we don t use an x-values to the left of zero. That is wh we could not put in - for x in the example, because - is not in the domain of = log x. Basic Characteristics of Logarithmic Graphs f(x) = log a x. The domain is (0, ).. The range is (-, ). 3. The x-intercept is (, 0). 4. The -axis is a vertical asmptote. 5. The function is increasing (a > 0). 6. The function is continuous. 7. The function passes the Horizontal Line test (ie. it is one-to-one) so it has an inverse function. 8. The function is a reflection of = a x over the line =x. 5
Section 3. Rigid Transformations Graph the following: = log x = log(x+) = log(x) = log(x+) = log(x) What kind of transformations do we have? = log(x+) is =log x shifted units to the left. = log(x) is = log x shifted unit down. What would =log(x+3) + 7 look like? =log(x+3) + 7 would be = log x shifted 3 units left and 7 units up. 6
Section 3. =log(x+3) + 7 Properties of Logarithms. log a = 0. log a a = 3. log a a x = x and a log ax = x 4. If log a x = log a, then x =. Examples: Solve the following equations: (a) log 5 x = log 5 8 (b) log 5 = x x = 8 5 x = x = 0 7
Section 3. c) log 7 x = 7 = x x = 7 Examples: Simplif the following: (a) log 6 6 x (b) 5 log 50 6 n = 6 x 5 log 50 = 0 n = x The Natural Logarithmic Function Remember f(x) = e x (where e.7) The inverse would be f(x) = log e x. Since this is used a great deal, we have a notation (and button on our calculator dedicated to it. Definition: The logarithmic function with base e is denoted f(x) = ln x *Remember that ln x is just logex. 8
Section 3. Properties of Natural Logarithms 5. ln = 0 6. ln e = 7. ln e x = x and e ln x = x 8. If ln x = ln, then x =. Examples: Evaluate the following: e (a) ln e 5 (b) e ln3 (c) ln e 5 3 ln e - = - Examples: Use our calculator to find the following. (a) ln 3 (b) ln 0. (c) ln (-) (d) ln(+ 5 ).099 -.609 Error.74 *Look at the graph of = ln x. Just as with = log x a, we cannot take the log of a negative number. = ln x 9
Section 3. Finding the Domain of the Logarithmic Function The domain of f(x) = log x and f(x) = ln x is (0, ) (a) What is the domain of f(x) = log(x+)? We must have x+ > 0 since what we take the log of must not be negative. Solving x+ > 0 we get x > -, so the domain is (-, ). *Note: This makes sense, because we know that f(x) = log (x+) is f(x) = log x shifted units left. The vertical asmptote of f(x) = log x is the -axis (x=0). The vertical asmptote of f(x) = log(x+) is x= -. = log(x+) 0
Section 3. (b) What is the domain of ln(3 - x)? We must have 3 x > 0 -x > -3 x < 3, so the domain is (-, 3) ln(3 x) (c) What is the domain of ln x? x >0, which means x > 0 or x < 0 (ie. x can be anthing but 0) So the domain is (-, 0) (0, ) ln x