Mathematics

Problem

Consider the logarithmic function

y=\log_a x

, where

a>0

and

a\neq1

. a. (i) State the domain and range of the function when

a>1

. b. (ii) Prove that the curve approaches the

y

-axis but never meets it, and hence obtain the equation of its vertical asymptote. c. (iii) Determine, with reasons, whether the curve is increasing or decreasing when (a)

a>1

, (b)

0<a<1

. d. (iv) On the same set of axes, sketch the graphs of

y=\log_2 x

and

y=\log_{1/3}x

, clearly labelling their common asymptote and the point through which both curves pass.

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