Solving $3^x + 4^x = 15$

Question

I'm trying to solve the problem, but I didn't get the way to do it:

$$3^x + 4^x = 15$$

I tried the $\ln$-way, but it didn't help. I'm wondering how to find $x$ in this case

Answer 1

The function on the left is an increasing function of $x$ and will have just one solution somewhere between $x=1$ and $x=2$. There is no way to "solve" the equation with a formula. You will need numerical methods.

Wolfram Alpha says the answer is about $x = 1.60057$.

Answer 2

As said in comments and answers, the is no analytical solution of the problem which means that you need some numerical method for solving the equation.

If you plot $$f(x)=3^x+4^x-15$$ it is very stiff while $$g(x)=\log(3^x+4^x)-\log(15)$$ looks very linear. This would help.

On the other side, you could ntice that $f(x)$ is bracketed by $2 \times 3^x-15$ and $2 \times 4^x-15$ which means that the solution is such that $$\frac{\log \left(\frac{15}{2}\right)}{\log (4)}<x <\frac{\log \left(\frac{15}{2}\right)}{\log (3)}$$ which are $1.45$ and $1.83$.

So, using Newton method, let us start at the midpoint of the interval; the successive iterates will be $$\left( \begin{array}{cc} n & x_n \\ 0 & 1.6437445 \\ 1 & 1.6005827 \\ 2 & 1.6005684 \end{array} \right)$$