Antilogarithm Inequality

[onemachine]

Here is the inequality

[TEX]\left( 1 - a \right) < 1[/TEX],

where [TEX] 0 < a < 1[/TEX]

Taking the base 2 antilogarithm of both sides gives

[TEX]2^{\left( 1 - a \right)} < 2[/TEX]

I am not sure if this is antilogarithm operation is allowed because the quanitity [TEX]\left( 1 - a \right)[/TEX] is less than one.

P.S. I am not trying to solve this for [TEX]a[/TEX]. I am just using this as an example so I can figure this out.

Thanks for your help!

[pka]

Here is the inequality

[TEX]\left( 1 - a \right) < 1[/TEX],
where [TEX] 0 < a < 1[/TEX]
Taking the base 2 antilogarithm of both sides gives
[TEX]2^{\left( 1 - a \right)} < 2[/TEX]
I am not sure if this is antilogarithm operation is allowed because the quanitity [TEX]\left( 1 - a \right)[/TEX] is less than one.
P.S. I am not trying to solve this for [TEX]a[/TEX]. I am just using this as an example so I can figure this out.

I not sure what antilogarithm means.

But [TEX]2^x[/TEX] is an everywhere increasing function.

Therefore for any [TEX]a<b[/TEX] you have [TEX]2^a<2^b[/TEX].