Open the “Logarithmic Functions” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Interactive Figures) or at bit.ly/3raFUGB.
(a) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of g(x) = log₂ (x − 5) is obtained from the graph of f (x) = log₂ x by shifting the graph of f (x) = log₂ x_______ (horizontally/ vertically), five units (right/left/up/down).

(b) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of g(x) = log₂ (x + 7) is obtained from the graph of f (x) = log₂ x by shifting the graph of f (x) = log₂ x______ (horizontally/ vertically), seven units_______ (right/left/up/down).

(c) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function and the vertical asymptote. Based on your observations, what would the vertical asymptote of f (x) = 2 log₂ (x − 5) be?

(d) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of k from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of g(x) = log₂ x + 5 is obtained from the graph of f (x) = log₂ x by shifting the graph of f (x) = log₂ x______ (horizontally/ vertically), five units______ (right/left/up/down).

(e) Set the values of a to 0.5, c to 1, h to 0, and k to 0. Check the “Show Inverse Function” box. Use the slider to adjust the value of h from −3 to 3 and pay attention to the vertical asymptote of the logarithmic function and the horizontal asymptote of the exponential function. If f (x) = 2 log3 (x + 2) − 1, what would the horizontal asymptote of the inverse exponential function be?

(f) If c < 0 and 0 < a < 1, the logarithmic function f (x) = c · log_a (x − h) + k is_______ (increasing/ decreasing) over its domain.

(g) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3 and pay attention to the equation of the logarithmic function. Note the domain for each value of h. Based on what you observe, what would be the domain of f (x) = 3 log₂( x − 5)? What would be the range of f ⁻¹? Express your answers in interval notation.