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

Question

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.

Answer 1

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?

Answer 2

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.