How to solve for exponent when adding fractions raised to unknown exponent?

Question

I'm sure this is probably an extremely simple problem but I'm stuck figuring this out.
For example:

$(\frac{1}{5})^{x} + (\frac{7}{10})^{x} = 1$

What would be the steps to solve for x?

Answer 1

This is a hard problem. There is no general closed form for $a^x + b^x = c$, and the best way is probably for you to use something like Newton's method to get an approximation. Here is a link to an answer outlining this

Answer 2

Both terms on the left side are decreasing functions in $x$. Note also that the left hand side is $2$ when $x=0$ and $0.9$ when $x=1$.

A binary search between $x=0$ and $x=1$ works. I'll show you how to do this semi-manually with Excel or some similar program.

Enter the following (column A is blank):

       A        B                         C
==================================================================
1               0                         =.2^B1 + .7^B1
2               1
3               =B2-A3*0.5*ABS(B2-B1)      

Fill down column C to Row 3, then fill down columns B and C down to about Row 40 or so, starting at Row 3.

Now, starting at Cell A3 and going down, enter $-1$ ("go down") if the value shown in column C is greater than $1$. Enter $1$ ("go up") if the value in column C is less than $1$.

By Cell A40, I found the answer to eleven significant figures: $0.83978030446$. (You may need to increase the number of significant figures shown in the cells to do this.)

Answer 3

As Brevan Ellefsen answered, Newton method as described in the link given is a very good candidate for solving this kind of equations.

Just for your curiosity, let us apply it to your case $$f(x)=(\frac{1}{5})^{x} + (\frac{7}{10})^{x} - 1$$ using $x_0=1$ since $f(0)=1$ and $f(1)=-\frac 1 {10}$. The successive iterates will then be $$x_1=0.825040253981992$$ $$x_2=0.839658576323164$$ $$x_3=0.839780296147214$$ $$x_4=0.839780304467821$$ which is the solution for fifteen significant figures.