Let be positive real numbers satisfying

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

Problem

Let

p,\,q,\,r

be positive real numbers satisfying

0<p<q<r

. Answer the following: (i) Compare the three numbers

\log_{5}p,\;\log_{5}q

and

\log_{5}r

, and write them in increasing order. (ii) Compare the three numbers

\log_{\frac14}p,\;\log_{\frac14}q

and

\log_{\frac14}r

, and write them in increasing order. (iii) Hence, or otherwise, prove that

\log_{5}q-\log_{\frac14}q\;\text{lies strictly between}\;\log_{5}p-\log_{\frac14}p\;\text{and}\;\log_{5}r-\log_{\frac14}r.