method of solving more complicated exponent problem

[galactus]

I would just like to see the different methods some may have of solving something like:

\(\displaystyle \L\\5^{x}+6^{x}=61\)

We can see the answer is x=2, but how to tackle it algebraically?.

Maybe I shouldn't have put this in Intermediate Algebra. If, perhaps, something like the Lambert function could be used.

 

[Denis]

Well, to get the ball rolling: 5^x < 61/2 and 6^x > 61/2 :idea:

 

[Deleted member 4993]

I would not even think about solving algebraically.

Straight go for graphical method - estimate answer. Then if more accuracy is required - go for numerical method - more specifically Newton-Raphson method.

 

[galactus]

Newton is how I went about it. I just wondered about other input.
I thought maybe someone would have something clever I haven't seen before.

 

[morson]

\(\displaystyle \L\ 5^x + 6^x = 5^2 + 6^2\)

So, x = 2.

 

[skeeter]

I would just like to see the different methods some may have of solving something like:

\(\displaystyle \L\\5^{x}+6^{x}=61\)

We can see the answer is x=2, but how to tackle it algebraically?.

 

[morson]

Yes, I was just showing that rewriting the right side is probably the most "algebraic" way of doing it, and also shows that it is the only solution.