exponential equation.

For discussing Olympiad Level Algebra (and Inequality) problems
jagdish
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Joined: Wed Jan 19, 2011 2:21 pm

exponential equation.

no. of real solution of the equation $4^x = x^2$
jagdish

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Re: exponential equation.

The given equation splits up into $2$ different equations.
$2^x=x$ for positive $x$. and $2^x=-x$ for negative $x$.
That means the equation we have to solve is this $2^x=|x|$
When $x$ is negative the left side is less than $1$ and tending to $0$.So no solution in this case.
If $x$ is positive the graph is a convex function starting from $0$.where the line $y=x$ is below the graph.So they never intersect.

*Mahi*
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Re: exponential equation.

zadid xcalibured wrote: When $x$ is negative the left side is less than $1$ and tending to $0$.So no solution in this case.
This statement is not quite right, there is one intersection, as RHS goes from $0$ to $\infty$ and LHS goes from $1$ to $0$ for $x= 0 \text{ to } -\infty$, and as both are strictly increasing/decreasing functions.

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Re: exponential equation.

How could i do this kind of shitty thing?????????? I don't know who gave me the right to turn a curve into a straight line. The first mistake is where i eliminated the square. But the problem is pretty easy.

*Mahi*
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Re: exponential equation.

zadid xcalibured wrote:I don't know who gave me the right to turn a curve into a straight line. The first mistake is where i eliminated the square.
I don't think that was a mistake, actually, it was a nice idea to eliminate the square.
Use $L^AT_EX$, It makes our work a lot easier!