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I stumbled upon this equation:

$$ log_2(x+4)=3-x $$

Naturally, I'd solve it graphically. But that got me wondering, can I also solve it algebraically and I honestly couldn't.

I tried rewriting it, so I could "see" where to go, but I didn't get anywhere.

$$ \frac{ln(x+4)}{ln(2)} = e^{ln(3-x)} $$

$$ 2^{3-x} = x+4 $$

Now, looking at similar posts, I noticed that everybody said, it is much easier to just graph it, but a question emerges:

Are there equations, which can not be solved algebraically, but only graphically?

and if it is possible to solve this algebraically, please let me know how to do it

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  • $\begingroup$ nearly all 1D equations can be ''solved'' graphically but nearly none can be solved algebraically $\endgroup$ – tired Sep 19 '16 at 14:33
  • $\begingroup$ Yes, there are equations, which cannot be solved algebracally. No, there are no equations which can be solved only graphically: there are at least three additional methods to solve, by guess or using one of numerical methods or using some special functions or methods designed for a specific kind of equations. $\endgroup$ – z100 Sep 19 '16 at 14:34
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    $\begingroup$ You can solve it with the Lambert W function: Write your equation as $2^x=8x+32$ and use the general case of en.wikipedia.org/wiki/Lambert_W_function#Example_1. $\endgroup$ – gammatester Sep 19 '16 at 14:35
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Hint: One can solve this using the so called Lambert W-function we have $$x=\frac{W(128 \log (2))-4 \log (2)}{\log (2)}$$

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  • $\begingroup$ With your $x$ I get $2^{x-3}-x-4 \approx -4.5411\;$ instead of $0$. The correct solutions is IMO $$x=-\frac{W(-\ln 2/128)+4\ln 2}{\ln2}$$ $\endgroup$ – gammatester Sep 19 '16 at 14:45
  • $\begingroup$ Maple 16 gives $$-{\frac {4\,\ln \left( 2 \right) -{\rm W} \left(128\,\ln \left( 2 \right) \right)}{\ln \left( 2 \right) }} $$ and this is the same as mine $\endgroup$ – Dr. Sonnhard Graubner Sep 19 '16 at 15:01
  • $\begingroup$ IMO, this doesn't actually answer the question... it just answers a specific case the OP didn't know about. $\endgroup$ – Simply Beautiful Art Sep 19 '16 at 15:02
  • $\begingroup$ this was not an answer but a comment can't you read it? $\endgroup$ – Dr. Sonnhard Graubner Sep 19 '16 at 15:05
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Sure, plenty of things we can't solve algebraically, which is why we do our best to approximate using approximation methods.

One such example is trying to solve $x^x=\Gamma(x)$, which can't be done with anything like the Lambert W function.

As a side note, what I would consider to be one of the simplest algebraic methods to finding solutions is fixed point iteration. For a simple example,

$$x^2=5$$

$$2x^2=5+x^2$$

$$x=\frac{5+x^2}{2x}$$

Suppose you think the solution is around $x=2$ (from graphing or guesswork), then see that

$$x\approx\frac{5+2^2}{2\times2}=\frac94$$

But if $x\approx\frac94$, then...

$$x\approx\frac{5+\left(\frac94\right)^2}{2\times\frac94}=\frac{161}{72}$$

etc.

$$x\approx\frac{5+\left(\frac{161}{72}\right)^2}{2\times\frac{161}{72}}=2.23606797792$$

The correct solution to $x^2=5$ is $x=\sqrt5=2.2360679775$.

And of course, there are other methods to doing this, but some require more than algebra.

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    $\begingroup$ As a sort of party trick, you can use this method to evaluate the square root of decently small numbers out a few decimals in your head. :D (Or other transcendental equations, if doable) $\endgroup$ – Simply Beautiful Art Sep 19 '16 at 14:59

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