Exponential Equation

What is the number of solutions to the equation $\large \sin(e^x) = 5^x + 5^{-x} ?$

#Geometry

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

$\sin(e^x)$ , because it is always the sine of a real number, has a maximum of 1 (whenever $e^x = \pi/2 + k\pi$ for some integer $k$ ), and $5^x+5^{-x}$ has its minimum at $x=0$ , where it is $2$ , so there are no solutions.

You can find that minimum with differential calculus, or in a simpler way. The function is obviously symmetric around the $y$ axis, so we can establish that it rises away from there as $x$ gets larger, and by symmetry we can be sure it rises in the other direction too. For $x>0$ , the function $5^x$ increases faster than $5^{-x}$ decreases, so the sum of those functions increases as $x$ does:

$5^{-(x+k)}-5^{-x}=\frac{1}{5^{x+k}}-\frac{1}{5^x} = \frac{5^x-5^{x+k}}{5^{2x+k}} = [5^{x+k}-5^x]\cdot\frac{-1}{5^{2x+k}}$

(And $5^{2x+k} > 1$ for $x>0$ .)

Mark C - 5 years, 1 month ago

Great observation!

Calvin Lin Staff - 5 years, 1 month ago

Thanks! It was actually not quite correct as I originally stated it, but I think it's better now.

Mark C - 5 years, 1 month ago

@Mark C – The easy way to find the minimum is to apply the arithmetic mean - geometric mean inequality.

$5^ x + 5 ^ {-x} \geq 2 \times \sqrt{ 5 ^ x \times 5 ^ {-x} } = 2$

Equality holds if $5 ^ x = 5 ^ {-x}$ , or when $x = 0$ .

Calvin Lin Staff - 5 years, 1 month ago

Let y = 5^x. So, 1/y = 5^(-x). Note that y, 1/y > 0. Thus, by AM-GM inequality y + 1/y > = 2.
Clearly, -1 =< sin(e^x) =< 1. Thus, sin(e^x) = 5^x + 5^(-x) is a contradiction. Therefore, no real value of x exist in this equation.

I am currently think for nonreal value of x.

Paul Ryan Longhas - 5 years, 1 month ago

When solving over the complex, there is almost always a solution. It's a result from complex analysis (Picard's theorem) that given any analytic function on the plane, the image misses at most one point.

Calvin Lin Staff - 5 years, 1 month ago

Go Here ,The easiest Way to Solve this Problem is by using A.M-G.M inequality

Sabhrant Sachan - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$