More concepts of Exponential function on inequality

More concepts of Exponential function on inequality

By abhijatindia - April 01, 2013

IMPORTANT CONCEPT

if ${a^b} > {a^c}$ , $a > 0,a \ne 1$

i) if

a > 1

,

$\Rightarrow b > c$

ii) If

a < 1

,

$\Rightarrow b < c$

Consider a exponential function, ${a^x}$

for, $a > 0,a \ne 1$

1. if

a > 1

let

a = 2

$y = {2^x}$

for,

x = - 1,0,1,2,3,4

$\begin{array}{l} {2^{ - 1}} = 1/2\\ {2^0} = 1\\ {2^1} = 2\\ {2^2} = 4 \end{array}$

2.if,

a < 1

let,

a = 1/2

$\begin{array}{l} {(1/2)^0} = 1\\ {(1/2)^1} = 1/2\\ {(1/2)^2} = 1/4 \end{array}$

for x = 0 the value is always 1 for exponential function,

and if

a > 1

, value of 'y' increases with increase in the value of 'x'.

means,

1. ${a^b} > {a^c} \Rightarrow b > c$ , for a > 1

a > 1

(and $a > 0,a \ne 1$ )

and if

a < 1

, value of 'y' decreases with increase in value of 'x'

2. ${a^b} > {a^c} \Rightarrow b < c$ , for a < 1

a < 1

( and $a > 0,a \ne 1$ )

from graph,

clearly from figure,

if ${a^b} > {a^c}$ , $a > 0,a \ne 1$

i) if

a > 1

,

$\Rightarrow b > c$

ii) If

a < 1

,

$\Rightarrow b < c$

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