Lesson 5.4 Exercises, pages

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1 Lesson 5.4 Eercises, pages 8 85 A 4. Evaluate each logarithm. a) log 4 6 b) log log c) log 6 6 d) log log Write each eponential epression as a logarithmic epression. a) 6 64 b) The base is. The logarithm is 6. So, 6 log 64 The base is 0. The logarithm is 4. So, 4 log 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 ii) log The base is 7. The eponent is 5. So, The base is. The eponent is. So, iii) log 0.0 iv) log a b The base is 0. The eponent is. So, The base is. The eponent is. So, Logarithms and the Logarithmic Function Solutions DO NOT COPY. P

2 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 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 is the power to which I raise 7 to get 6 807, which is 5. That is, , 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 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

3 . 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 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 Logarithms and the Logarithmic Function Solutions DO NOT COPY. P

4 . 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 So, log. Identify powers of close to and 7 8 log 6 < log 00< log 7 So, 6< log 00< 7 An estimate is: log So, log 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 y 8 8 y log 8 From the table, the functions are inverses 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 and 8 So, < log 6.5< An estimate is: log So, log Identify powers of close to.8. 0 and So, 0< log.8< An estimate is: log So, log P DO NOT COPY. 5.4 Logarithms and the Logarithmic Function Solutions

5 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 y log y 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 Logarithms and the Logarithmic Function Solutions DO NOT COPY. P