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Is it possible to solve for $$3^x=3-x$$ without graphing it?

This question is in the section: Solving Exponential equations in my math textbook.

I have tried to log both sides and solve for it however that just leads me back to the original equation.

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  • $\begingroup$ Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand. $\endgroup$
    – warjwar8
    Commented Nov 19, 2018 at 9:11
  • $\begingroup$ Have you tried to solve $3^x + x = 3$? $\endgroup$ Commented Nov 19, 2018 at 16:31

2 Answers 2

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HINT

Let consider

$$f(x)=3^x-(3-x)$$

and note that

  • $f(0)=-2$
  • $f(1)=1$

therefore by IVT there is a solution in $(0,1)$.

Can you show that this is the unique solution?

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  • $\begingroup$ If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number. $\endgroup$
    – jayant98
    Commented Nov 19, 2018 at 8:49
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    $\begingroup$ @jayant98 We can consider the sign of $f'(x)=3^x \log 3 +1$. $\endgroup$
    – user
    Commented Nov 19, 2018 at 8:56
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    $\begingroup$ Yes Sir! Another way which is more simple and effective. $\endgroup$
    – jayant98
    Commented Nov 19, 2018 at 8:58
  • $\begingroup$ @jayant98 Call me simply gimusi! Bye $\endgroup$
    – user
    Commented Nov 19, 2018 at 9:03
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Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x \approx 0.742$ but cannot give you an exact value.

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