Let be a real number such that and .

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Let

a

be a real number such that

a>0

and

a\neq 1

. (i) Starting from the equation

a^{x}=y

with

y>0

, derive an expression for

x

in terms of

y

using logarithms. Give clear reasons for each algebraic step. (ii) Using your result from part (i), determine the value of

x

when

a=5

and

y=125

. (iii) Treating

x

as a function of

y

, explain why the graphs of

y=a^{x}

and

x=\log_{a}y

are reflections of each other in the line

y=x

. Support your explanation with an example point on each graph.