Calc Real $x$ in Exponential Equation

[juantheron]

Calculation of all real values of $x$ for which $3^x+6^x = 4^x+5^x$

 

[chisigma]

Re: exponential equation

Calculation of all real values of $x$ for which $3^x+6^x = 4^x+5^x$

[sp]Let's consider the function...

$\displaystyle f(x) = 3^{x} - 4^{x} - 5^{x} + 6^{x}\ (1)$

It is immediate to realize that f(x) vanishes in x=0 and x=1. The derivative in x=0 is...

$\displaystyle f' (0) = ln \frac{9}{10}< 0\ (2)$

... so that around x=0 is f(x)>0 for x<0 and vice versa. If x is negative then the term $3^{x}$ is dominating and is f(x)>0, so that no negative roots exist. For x>1 the term $6^{x}$ is dominating so that there is no roots greater than 1. The conclusion is that x=0 and x=1 are the only roots...[/sp]

Kind regards

$\chi$ $\sigma$