Can you solve this exponential equation?

Question

I'm supposed to solve the equation $$3^x + 3^\sqrt x = 90$$

What steps do I need in order to get the solution $x=4$?

there is no nice method. The left hand side increases with (positive) $x,$ you can see that $x=4$ works, so that is it. If they replaced $90$ with either $89$ or $91$ you would be out of luck. — Will Jagy, Oct 07 '14 at 01:46
Notice that for large x, the first term dominates the second, so the answer is approximately log3 of the right-hand side. — user541686, Oct 07 '14 at 07:23
Two people have already voted to close this question stating the reason: This question is missing context or other details. Perhaps you could mention where you encountered this equation or why you are interested in it. (I.e., add some context, so that the question will not be closed.) — Martin Sleziak, Oct 07 '14 at 07:28

score 3 · Answer 1 · answered Oct 07 '14 at 01:46

There is in truth, no rigorous "algebraic" way to solve this.

The only real viable way is to "guess and check" or "spot" the integral solution. Alternatively, you can use iterative methods to approximate a solution, which will give you a value "very close" to $4$ - upon which you can test $4$ and find that it solves the equation exactly. You can then use curve sketching or some other means to show that it's the only solution as the function $f(x) = 3^x + 3^{\sqrt x}$ is strictly increasing.

score 2 · Answer 2 · answered Oct 07 '14 at 03:35

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Another way is by inspection. Since the left hand side is sums of powers of 3, can we write 90 as the sum of powers of 3?

Yes, $3^x+3^\sqrt{x}=90=81+9=3^4+3^2.$

Thus $x=4$.

answered Oct 07 '14 at 03:35

Laars Helenius

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Paul · Answer 3 · 2014-10-07T13:08:04.553

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I cannot find a better way then the following to solve it. We may using by observing that $x=4$ is the root of this equation and by observing that the function $f(x)= 3^x + 3^\sqrt x$ is an increasing function, which implies $x=4$ is the unique root.

May it helps!

edited Oct 07 '14 at 13:08

answered Oct 07 '14 at 01:20

Paul

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how it comes 3^(log3t)^2= 9t ? – azazy Oct 07 '14 at 01:29
(log x )^2 not = 2 log x – azazy Oct 07 '14 at 01:30
It has somththing wrong. Sorry @azazy. Let me think for a moment. If I have a right way to solve i will edited, otherwise I well delete this post. – Paul Oct 07 '14 at 01:33
never mind take your time but dont delete the post plz – azazy Oct 07 '14 at 01:36
This is incorrect. $3^{(\log_3t)^2} \neq 9t$. It can only be "simplified" to $t^{\log_3 t} = t^{\sqrt x}$, which doesn't get you very far. – Deepak Oct 07 '14 at 01:40
@Alan That's wrong also. – Deepak Oct 07 '14 at 01:43
True, strange. Deleting comment – Alan Oct 07 '14 at 01:44
Escape is not the solution – azazy Oct 07 '14 at 01:59
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@azazy: Ya. However, not every question has a nice way to be solved. This happens not only in maths but also in the man living. – Paul Oct 07 '14 at 02:04

Can you solve this exponential equation?

3 Answers3