Lesson 5.4 Eercises, pages 8 85 A 4. Evaluate each logarithm. a) log 4 6 b) log 00 000 4 log 0 0 5 5 c) log 6 6 d) log log 6 6 4 4 5. Write each eponential epression as a logarithmic epression. a) 6 64 b) 0 4 0 000 The base is. The logarithm is 6. So, 6 log 64 The base is 0. The logarithm is 4. So, 4 log 0 000 c) 4 - d) The base is 4. The logarithm is The base is.. The logarithm is. So, a b So, log 6. a) Write each logarithmic epression as an eponential epression. i) log 7 6 807 5 ii) log The base is 7. The eponent is 5. So, 6 807 7 5 The base is. The eponent is. So, iii) log 0.0 iv) log a b The base is 0. The eponent is. So, 0.0 0 The base is. The eponent is. So, 0 5.4 Logarithms and the Logarithmic Function Solutions DO NOT COPY. P
b) Use one pair of statements from part a to eplain the relationship between a logarithmic epression and an eponential epression. Sample response: I start with this logarithmic epression: log 7 6 807 5. The logarithm of a number is the power to which the base of the logarithm is raised to get the number. So, the logarithm base 7 of 6 807 is the power to which I raise 7 to get 6 807, which is 5. That is, 6 807 7 5, which is the equivalent eponential epression. B 7. Evaluate each logarithm. a) log 4 8 b) log a 7 b ( ) log A 4 B log A B 4 log A B c) log 6 d) log 4 4 log 6 6 0 0 log A # 4B log A 4B 4 8. a) Write as a logarithm b) Write as a logarithm with base. with base. Use: n log b b n log So, log 8 Use: n log b b n log So, log. Determine the value of log b. Justify the answer. log b is the eponent of base b, when b is raised to a power and the result is. The only way to get is to raise the base to the power 0. So, log b 0 0. How are the domain and range of the functions y b and y log b related? Since the functions y b and y log b are inverse functions, the domain of y b is the range of y log b, and the range of y b is the domain of y log b P DO NOT COPY. 5.4 Logarithms and the Logarithmic Function Solutions
. a) Use a table of values to graph y log 5 Determine values for y 5, then interchange the coordinates for the table of values for y log 5 5 y log 5 0 y y log 5 0 4 8 6 0 4 5 5 b) Identify the intercepts, the equation of the asymptote, the domain, and the range of the function. The graph does not intersect the y-ais, so it does not have a y-intercept. The graph has -intercept. The y-ais is a vertical asymptote; its equation is 0. The domain of the function is > 0. The range of the function is y ç. c) What is the significance of the asymptote? The asymptote signifies that the logarithm of any positive number very close to 0 eists, but the logarithm of 0 does not eist. The graph approaches the line 0 but never intersects it.. a) Use technology to graph y log Identify the intercepts, the equation of the asymptote, the domain, and the range of the function. On the Y screen, input Y log (X), then press: s. From the table of values, or the CALC feature, there is no y-intercept, the -intercept is, and the equation of the asymptote is 0. The domain is > 0; the range is y ç. b) What is the equation of the inverse of y log? y log can be written y log 0 Interchange and y, then solve for y. log 0 y Write the equivalent eponential statement. y 0 5.4 Logarithms and the Logarithmic Function Solutions DO NOT COPY. P
. Use benchmarks to estimate the value of each logarithm, to the nearest tenth. a) log b) log 00 Identify powers of close to. and 7 log < log < log So, < log < An estimate is: log...57846..5505 So, log. Identify powers of close to 00. 6 64 and 7 8 log 6 < log 00< log 7 So, 6< log 00< 7 An estimate is: log 00 6.6 6.6 7.0058606 6.7 0.68067 So, log 00 6.6 4. Write the equations of an eponential function and a logarithmic function with the same base. Use graphs of these functions to demonstrate that each function is the inverse of the other. Sample response: Here are the tables of values and graphs of y 8 and y log 8 Plot the points, then join them with smooth curves. y 8 y log 8 8 y y 8 0.5 0 0.5 0 6 4 y 8 8 y log 8 From the table, the functions are inverses 0 4 6 8 because the coordinates of corresponding points are interchanged. From the graph, the functions are inverses because their graphs are reflections of each other in the line y 5. Use benchmarks to estimate the value of each logarithm to the nearest tenth. a) log 6.5 b) log.8 Identify powers of close to 6.5. 4 and 8 So, < log 6.5< An estimate is: log 6.5.7.7 6.4807.8 6.64404506 So, log 6.5.7 Identify powers of close to.8. 0 and So, 0< log.8< An estimate is: log.8 0.6 0.6.8045 0.5.7050808 So, log.8 0.5 P DO NOT COPY. 5.4 Logarithms and the Logarithmic Function Solutions
C 6. Graph y = log How is the graph of this function related to the graph of y log? Make a table of values for y a, then interchange the coordinates b for y log Plot the points, then join them with a smooth curve. y a b 8 4 0 0.5 0.5 y 8 4 0 0.5 0.5 log y 0 4 6 8 y log I looked at the graph of y log that I drew on page 75. The graph of y log is the reflection of the graph of y log in the -ais. 7. On a graphing calculator, the key μ calculates the value of a logarithm whose base is the irrational number e. The number e is known as Euler s constant. Logarithms with base e are called natural logarithms. a) Graph y ln Sketch the graph. Input: Y μ (X), then press: s b) Determine the value of e to the nearest thousandth. When y, log e, so e, or e On the screen in part a, graph Y, then determine the approimate -coordinate of the point of intersection:.7888 The value of e is approimately.78. 4 5.4 Logarithms and the Logarithmic Function Solutions DO NOT COPY. P