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January 10th, 2015, 04:44 PM   #1
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Find me the solution...

Find me the solution. No iterative methods. No cheap guesswork.

Isolate x.

$\displaystyle { 2 }^{ x }+{ 3 }^{ x }=97$

Blow my mind if you have to.
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January 10th, 2015, 05:36 PM   #2
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The unique solution is x = 4. The LHS is strictly increasing and the RHS is constant.
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January 11th, 2015, 01:20 AM   #3
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The unique solution is x=4, but I'd like to know how to isolate x and get the answer using algebraic methods.

Isn't it possible?
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January 11th, 2015, 03:55 AM   #4
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$\displaystyle f(x) = { 2 }^{ x }+{ 3 }^{ x }$ is continuous. So there is only one real root for $\displaystyle f(x) = a$, $\displaystyle a \in \mathbb{R}$.

$\displaystyle g(x)=2^x$ & $\displaystyle h(x)=97-3^x$ has one intersection point, x=4.
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January 11th, 2015, 10:00 AM   #5
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One can't isolate $x$ algebraically.
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January 11th, 2015, 01:59 PM   #6
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This is depressing.

What would happen if I created a similar problem with more complexity?

Would we just have to iterate?

Bizarre how we haven't found a way to solve these problems algebraically.

Will it happen in my lifetime?

------------------------------

One more question...

There are obviously mathematical problems that can only be solved using computers.

How do we program computers to solve such problems?

Do we just teach them how to iterate?

Last edited by perfect_world; January 11th, 2015 at 02:02 PM.
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January 11th, 2015, 04:01 PM   #7
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Let's consider instead a simpler problem: $2^x= 97$. Its real solution is $x = \log_2(97)$, but how would you evaluate that? For convenience, you would probably use a computer, and the computer program would probably use... iteration! Unfortunately, the resulting value would still be slightly inaccurate (due to rounding).
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January 12th, 2015, 02:54 AM   #8
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I suppose the same would be true about calculating, let's say tan(25) or arcsin(0.45)?

If a computing machine were to come close to a number (for instance 2 root 2), it would spit out 2 root 2, but we wouldn't be able to spot such patterns as quickly as a machine as such.

Frustrates me how graphing calculator apps can get these answers in a flash, whilst us humans have to do such tedious work.

I'd really like to know how to come up with accurate answers to these problems using iterative methods. I know I'll have to use computers when solving the most complex problems, but to understand how a computer does its calculations would give me a lot of satisfaction.

At least I'd be able to say I understand how they operate. I certainly don't expect to calculate as well as a computer though. I'm not that deluded.

-BTW, I personally think that we may one day be able to answer such problems using a new form of digital mathematics (very simple) and no computers. I'm just theorising though...

Last edited by perfect_world; January 12th, 2015 at 03:12 AM.
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January 12th, 2015, 04:25 AM   #9
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Quote:
Originally Posted by perfect_world View Post
I'd really like to know how to come up with accurate answers to these problems using iterative methods. I know I'll have to use computers when solving the most complex problems, but to understand how a computer does its calculations would give me a lot of satisfaction.
This is covered in numerical analysis. The basic idea is that there is an approximating formula for different regions of the function, and then a function is iterated which replaces an estimate with a closer estimate. The good functions roughly double the precision at each step.

In practice some functions are decomposed; for example, 6^x would typically be calculated as (something like) exp(x * log(6)). The actual code is typically highly complex, but the basic idea is not hard.
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