0

Find $x$: $$9^x = 4^x + 6^x$$

This was in my exam today, and I have no idea, would really help if someone taught me, I would love to know how to write math on this website and is there a way to like search a topic for a 10th grader where i can find many questions and solve them?

Blue
  • 75,673
memeguy
  • 47
  • 3
    The general question of solving for $x$ in $a^x+b^x=c^x$ where $a,b,c$ are just arbitrary numbers like $3^x+5^x=7^x$ or $11^x+11^x=12^x$ or other examples... this general question is not solvable using "simple" or "elementary" methods. As a tenth grader you are not expected to be able to solve random problems like this. Even experts would need to rely on things like the Lambert-W function (a non-elementary function which is particularly ugly and difficult to use in practice) or numerical methods. – JMoravitz Sep 19 '23 at 12:56
  • 2
    @JMoravitz Actually, this case is transformable into a quadratic. – Deepak Sep 19 '23 at 12:59
  • 1
    If you were given a problem like this, it was either a mistake, or they expected you to be able to find it by inspection or some special property specific to these particular numbers. We can tell right away that there should be an answer since both functions are monotonic increasing and at $x=1$ you have $4^x+6^x=4^1+6^1=10$ while $9^x=9^1=9$ is less than that, while later at $x=2$ you have $4^2+6^2=16+36=52$ while $9^2=81$ is more than that, so the answer should be somewhere between $1$ and $2$, but that the answer can't be a whole number since left would be odd while right is even. – JMoravitz Sep 19 '23 at 12:59
  • 6
    @JMoravitz Because $6$ is the geometric mean of $4$ and $9$, this particular case does have a solution for example if you let $y=\left(\frac32\right)^x$. Familiarity with logarithms might help express this solution – Henry Sep 19 '23 at 12:59
  • 1
    @JMoravitz Thank you sir, just know that I appreciate you and aspire to become like you(very good in maths and physics) one day – memeguy Sep 19 '23 at 13:02
  • @Henry Sir can you maybe provide the solution? And also can you tell me what is a monotonic function? I know logarithms I did it in class 9th. – memeguy Sep 19 '23 at 13:03
  • monotonic (increasing) just means that it only grows and doesn't ever turn around. – JMoravitz Sep 19 '23 at 13:04
  • @JMoravitz thank u sir – memeguy Sep 19 '23 at 13:04
  • As for your later questions , the answer is Yes. You can find questions (and read nice answers) by searching them under appropriate tags of your convenient level. – An_Elephant Sep 20 '23 at 07:13
  • See https://math.stackexchange.com/q/3625366/42969 and the linked questions – found with Approach0 – Martin R Sep 20 '23 at 09:12
  • 1
    See also How to search on this site?, in particular How to search with Approach0. – Martin R Sep 20 '23 at 09:13

1 Answers1

37

$$9^x = 4^x + 6^x$$

Dividing by $4^x$ , we get :

$$\left(\frac{9}{4}\right)^x = 1 + \left(\frac{3}{2}\right)^x$$

$$\left(\frac{3}{2}\right)^{2x} = 1 + \left(\frac{3}{2}\right)^{x}$$

Substituting $\left(\frac{3}{2}\right)^x = t $ , you get a quadratic equation in $t$, $$t^2 = 1 + t$$

Now, solve for $t$ and then solve for $x$.

An_Elephant
  • 2,704
Motivix
  • 481