MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions
Problem
Consider the two functions f(x)=log3x and g(x)=log1/4x.
(i) Without performing any calculation of logarithms, explain which of the two functions is increasing on (0,∞) and which is decreasing. Give a brief reason based on the value of the base.
(ii) Describe the concavity (concave up or concave down) of each graph and justify your answer conceptually.
(iii) Sketch both graphs on the same set of axes, marking clearly the common intercept on the x-axis and the common vertical asymptote.
(iv) Using your sketch, compare the numerical values of f(2) and g(2) and state which is larger. Give a short verbal justification referring to the shape of each curve.