I use Mathematica to solve this equation $$8 \left(3^x+5^x+7^x\right)=5\cdot 2^x+2\cdot 4^x+17\cdot 6^x$$ and get three solutions $x=0\lor x=1\lor x=2.$ I don't know how to solve by hand. How can I solve it?
Asked
Active
Viewed 74 times
-1
-
Let $f(x) = 8 \left(3^x+5^x+7^x\right) -(5\cdot 2^x+2\cdot 4^x+17\cdot 6^x)$, use caculus to draw the graph of $f(x)$ and show that $x=0, 1, 2$ are only solutions of $f(x)=0$. – Basics Dec 18 '23 at 11:04
-
Draw the graph, I think not a solve. – John Paul Peter Dec 18 '23 at 14:39
-
@John, what did you try in order to solve your exercise ? Read this thread and learn how to ask a good question on this site. Your post is being closed by moderators because it is a very low-quality question. – Angelo Dec 18 '23 at 16:08
1 Answers
2
I'll assume you're searching solutions in $\mathbb{Z}$. Now it'easy to see that if $x$ is negative, let's say $x=-q$ with $q$ positive, the equation is not solvable as we can multiply both sides by $3^q5^q7^q$, so that the left side is an integer, but the other one clearly is not. On the other hand if $x$ is greater than $3$, we get that the greatest power of 2 dividing the left side is always $8$, whereas the right one is divisible by $2^x$. So you just need to check for the other cases.
temp
- 105
- 1
- 6