Consider the base

MATHEMATICS · Subjective · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Consider the base

b=2

. (i) State the property that relates the order of real numbers

x_{1}

and

x_{2}

to the order of the values

2^{x_{1}}

and

2^{x_{2}}

when

b>1

. (ii) Using only this property, arrange the numbers

2^{-3}

2^{\sqrt{2}}

and

2^{5/2}

in ascending order. (iii) Give a brief justification for the chain of inequalities

2^{-4}<2^{-1}<1<2^{1/2}.