Consider the logarithmic function where and .

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

Problem

Consider the logarithmic function

y=\log_{a}x

where

a>0

and

a\neq1

. (i) Show that the expression

\log_{a}x

is not defined for

x\le0

. (ii) Using the fact that

y=a^{x}

is the inverse of

y=\log_{a}x

, explain why the range of

y=\log_{a}x

is the entire set of real numbers. (iii) Describe how the key features of the graph of

y=\log_{a}x

support your conclusions in parts (i) and (ii).