Daniel Dalton <daniel.dalton47 at gmail.com> writes:
> Hello,
>
> I need to solve the following.
>
> 3^x=x+4
>
> My text book tells me to graph it and see where the graphs (y=x+4, and
> y=3^x) meet. Unfortunately, I'm blind and can't read very accurately off
> the graph, and thus am using maxima instead of the recommended devices
> by my school.
>
> I tried this:
> solve([y=x+4, y=3^x],[x,y]);
>
> Obtaining this:
>
> algsys: tried and failed to reduce system to a polynomial in one variable; give up.
> -- an error. To debug this try: debugmode(true);
>
> Can anyone recommend a way I should solve this question with maxima?
Hi,
The reason that your textbook recommends solving this by looking at a
graph is that the equation isn't easy to solve symbolically. (Indeed, I
suspect that there isn't a solution that can be written in elementary
functions, but I'm not sure how to prove that).
As such, you're going to have to use some numerical method (staring at a
graph basically gives you low-tech numerical root finding). Maxima has a
function called find_root that will do the job for you nicely, but it
needs to know where to look and it only copes with functions of one
variable.
Firstly, we can turn this into a problem of only one variable since
y=x+4 means that x=y-4 so y=3^x gives us y = 3^(y-4). Functions like
find_root actually want an expression that they can test for zero, so
let's look at y - 3^(y-4).
Now we need to know where to look for a root. Notice that if y < 0 the
expression must be negative since 3^(anything) is positive and is being
subtracted from a negative number. If y = 1, the expression is
definitely positive, since 3^(-3) < 1. If y is large, the expression
will be negative since 3^(y-4) grows exponentially.
As a result (since the expression gives a continuous function of y), we
know there must be at least two roots. Indeed, there really are only two
roots, which you could check by differentiating (then the expression
becomes easier to solve, because the "y" term becomes constant and you
can take logarithms).
Now we can finally use find_root:
(%i18) find_root (y - 3^(y-4), y, 1, 10);
(%o18) 5.56191876632454
(%i19) find_root (y - 3^(y-4), y, 0, 1);
(%o19) .01251661588402949
Tada!
Incidentally, I suspect that was slightly more information than your
textbook expected you to need. In particularly, if you can look at the
graph, it becomes "obvious" that there are two roots. Of course, for
weirder functions, even those with perfect eyesight would need to use
this sort of argument...
Rupert
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