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# Find values of x satisfying equations with logarithms

Find an such that the following equations hold.

1. .
2. .
3. for .
4. .

1. We compute,

2. We compute,

3. We compute, for ,

4. We compute,

1. Eiji says:

For c, How do you deduce x=4 from x^2 = 2^x?

• Artem says:

I am also interested if it can be solved analytically.
But I think here the method of “informed guess” was taken – we just guessed the number. I personally checked the graph plotter to see this.

• TRC YX says:

I searched on WolframAlpha, and the solution is analytically represented by a lambert function, where is the inversion of , so that

and it’s impossible to express this in terms of elementary functions.

The equation can be separated into two equations and . Since is only defined for positive values, we can only solve the first one (while the second one does yield a negative real solution).

Since multiple can have the same , so there are two branches of , and hence two solutions of the above equation, which are and respectively, and is the answer we need here.

However I’m not sure how to look through the expression of the solution to see and , other than guessing the value of . (

• TRC YX says:

Sorry that there’s some problem with my latex. I hope this works.