Name Student ID Number Group Name Group Members Logarithmic Functions 1. Solve the equations below. xx = xx = 5. Were you able solve both equations above? If so, was one of the equations easier to solve than the other? Why? 3. Solve the equations below. xx = 3 xx = 30. Were you able solve both equations above? If so, was one of the equations easier to solve than the other? Why?
Consider the function ff(xx) = xx. If we attempt to find the inverse function (is f one-to-one?), we see: Now switching x and y: ff(xx) = xx yy = xx xx = yy From here, how can we solve for y? The answer is we can t. We need to define a new function, a logarithmic function, which allows us to undo an exponential function. If xx and bb are positive real numbers such that bb 1, then yy = log bb xx is called the logarithmic function base bb, where yy = log bb xx xx = bb yy Notes: The value yy is the exponent to which bb must be raised to obtain xx. The value of yy is called the logarithm, bb is called the base, and xx is called the argument. The equations yy = log bb xx and xx = bb yy both define the same relationship between xx and yy. The expression yy = log bb xx is called the logarithmic form and xx = bb yy is called the exponential form. In addition to the general form of a logarithm as shown above, we have special notation for two of the logarithms. The natural logarithm function is denoted ff(xx) = ln xx. The common logarithm function is denoted ff(xx) = log xx. 5. Write each equation in exponential form: a. log 3 9 = b. log 00 = 3. Write each equation in logarithmic form: a. 5 = 5 b. 0 = 1 000
7. Evaluate each logarithmic expression. a. log b. log 3 1 c. ln ee 9 d. log 5 ( 5) e. log 3 1 1 f. log 0 g. log h. 5ln ee i. 3log 3 j. log 1 k. log 0 l. log 1 5 m. log 7 9 5 n. log
Basic Logarithmic Properties For bb > 0, bb 1: log bb 1 = log bb bb = bb log bb xx = log bb bb xx = **Make sure to say: log base b of 1 log base b of b b to the power log base b of x log base b of b to the x. Recall the graphs of ff(xx) = xx, gg(xx) = 3 xx, h(xx) = xx, and jj(xx) = ee xx. Use them to plot the inverses ff 1 (xx) = log xx, gg 1 (xx) = log 3 xx, h 1 (xx) = log xx, and jj 1 (xx) = ln xx. f(x) = x g(x) = 3 x 5 5 5 5 h(x) = x j(x) = e x 5 5 5 5
9. Recalling that yy = log bb xx xx = bb yy and referencing the graphs you just created, are there any values of x that we cannot use? In other words, what is the domain of a logarithm of the form yy = log bb xx?. What is the range of a logarithm of the form yy = log bb xx? 11. What is the connection between the domain and range of the exponential functions versus the logarithmic functions? Recall the properties of the graphs of exponential functions: The graph of an exponential function ff(xx) = bb xx has the following properties. 1. If bb > 1, ff is an increasing exponential function (exponential growth function). If 0 < bb < 1, ff is a decreasing exponential function (exponential decay function).. The domain is the set of all real numbers, (, ). 3. The range is (0, ).. The line yy = 0 is a horizontal asymptote. 5. The function passes through the point (0, 1) because ff(0) = bb 0 = 1. We will unofficially call this the pivot point. Compare to the properties of the graphs of logarithmic functions: The graph of a logarithmic function ff(xx) = log bb xx has the following properties. 1. If bb > 1, ff is an increasing logarithmic function (logarithmic growth function). If 0 < bb < 1, ff is a decreasing logarithmic function (logarithmic decay function).. The domain is (0, ). 3. The range is the set of all real numbers, (, ).. The line xx = 0 is a vertical asymptote. 5. The function passes through the point (1, 0) because ff(0) = log bb 1 = 0. We will unofficially call this the pivot point.
1. Graph ff(xx) = log xx and gg(xx) = log (xx + ) 1 on the same set of axes. List the transformations you would apply to ff(xx) to get gg(xx) and label the intercepts, pivot point, and asymptote. Transformations: 5 5 13. Graph ff(xx) = log 3 xx and gg(xx) = log 3 xx on the same set of axes. List the transformations you would apply to ff(xx) to get gg(xx) and label the intercepts, pivot point, and asymptote. Transformations: 5 5
1. Graph ff(xx) = log 3 xx and gg(xx) = log 3 (xx 1) + 3 on the same set of axes. List the transformations you would apply to ff(xx) to get gg(xx) and label the intercepts, pivot point, and asymptote. Transformations: 5 5 15. In number 9, you should have determined that the domain of a function of the form ff(xx) = log bb xx is (0, ). If there are horizontal transformations, the domain will change. To determine the domain of any logarithmic function, the argument of the logarithm must be positive. Use this information to determine the domain of each of the following functions. Write your answer in interval notation. a. ff(xx) = log 5 (xx + ) b. pp(xx) = ln( xx)
c. ff(xx) = log (3xx 7) + d. qq(xx) = log (xx 5xx 1) e. rr(xx) = log xx 5xx 1 + xx+1 xx