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I try to solve the inequality

x - 1 - (Log[x])/(Log[2]) < (1/Log[2]) Log[(1 + (Log[10^6] + Log[x] + x*Log[x/2])/Log[a])] 

where a:=3+2*Sqrt[2].

The commands Reduce and NSolve do not work on my computer.

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  • $\begingroup$ The message from Reduce says it all: This system cannot be solved with the methods available to Reduce. I doubt there's an analytic solution; for a numerical one, see my answer below. $\endgroup$ – corey979 Sep 6 '17 at 13:46
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Define for simplicity

a = 3 + 2*Sqrt[2];
f[x_] := x - 1 - Log[x]/Log[2]
g[x_] := Log[1 + (Log[10^6] + Log[x] + x Log[x/2])/Log[a]]/Log[2]

Then

Plot[{f[x], g[x]}, {x, 0, 10}, PlotLabels -> "Expressions"]

enter image description here

So the intersections are at

FindRoot[f[x] == g[x], {x, 0.1}]

{x -> 0.0728923}

and

FindRoot[f[x] == g[x], {x, 8}]

{x -> 8.03971}

Because you want to solve

$$f(x) < g(x)$$

then the solution is

$$x\in(0.0728923; 8.03971).$$

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  • $\begingroup$ You could a domain restricted Solve as well: Solve[f[x] == g[x] && -10 < x < 100, x] will produce the intersection points. $\endgroup$ – Carl Woll Sep 6 '17 at 14:56

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