A logarithmic function has vertical asymptote and passes through the points and

MATHEMATICS · Subjective · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

A logarithmic function has vertical asymptote

x=1

and passes through the points

(2,2)

and

(5,3)

. (i) Assuming the equation is of the form

y=\log_{a}(x-1)+c

, determine the base

a

and the constant

c

. (ii) Write the equation of the inverse function and simplify it so that it is expressed as an exponential function of the form

y=\alpha^{x}+\beta

. (iii) Using your inverse function, find the

x

-coordinate of the point on the exponential curve where the ordinate (

y

-value ) equals

1