Consider the logarithmic function

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Consider the logarithmic function y=log⁡axy=\log_a x, where a>0a>0 and a≠1a\neq1. (i) State the domain and range of the function when a>1a>1. (ii) Prove that the curve approaches the yy-axis but never meets it, and hence obtain the equation of its vertical asymptote. (iii) Determine, with reasons, whether the curve is increasing or decreasing when (a) a>1a>1, (b) 0<a<10<a<1. (iv) On the same set of axes, sketch the graphs of y=log⁡2xy=\log_2 x and y=log⁡1/3xy=\log_{1/3}x, clearly labelling their common asymptote and the point through which both curves pass.

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