More concepts of Exponential function on inequality

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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 (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 ( 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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