# How to solve $3^x=3-x$

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.

• 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. Commented Nov 19, 2018 at 9:11
• Have you tried to solve $3^x + x = 3$? Commented Nov 19, 2018 at 16:31

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?

• 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. Commented Nov 19, 2018 at 8:49
• @jayant98 We can consider the sign of $f'(x)=3^x \log 3 +1$.
– user
Commented Nov 19, 2018 at 8:56
• Yes Sir! Another way which is more simple and effective. Commented Nov 19, 2018 at 8:58
• @jayant98 Call me simply gimusi! Bye
– user
Commented Nov 19, 2018 at 9:03

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.