Solving $3^x + 4^x = 15$ 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
 A: 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 $x = 1.60057$.
A: 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
$$\left(
\begin{array}{cc}
 n & x_n \\
 0 & 1.6437445 \\
 1 & 1.6005827 \\
 2 & 1.6005684
\end{array}
\right)$$
