To solve the equation [tex]\(x + 5 = -3^x + 4\)[/tex], we will proceed through several steps. The aim is to isolate [tex]\(x\)[/tex] and find its value.
1. Rewrite the Equation:
[tex]\[ x + 5 = -3^x + 4 \][/tex]
2. Simplify the Equation:
Let's get all terms involving [tex]\(x\)[/tex] on one side and constant terms on the other side:
[tex]\[ x + 5 - 4 = -3^x \][/tex]
This simplifies to:
[tex]\[ x + 1 = -3^x \][/tex]
3. Isolate the Exponential Term:
[tex]\[ x + 1 = -3^x \][/tex]
4. Introduce the Lambert W Function:
The Lambert W function, [tex]\(W(z)\)[/tex], is defined as the inverse function of [tex]\( f(W) = W e^W \)[/tex], meaning [tex]\( W e^W = z \)[/tex].
Here, we rewrite the equation to apply the Lambert W function:
[tex]\[ x + 1 = -3^x \][/tex]
We know that solving equations involving variables both inside and outside of an exponential function typically involves the Lambert W function. We recognize that this equation can't be simplified by elementary algebraic methods alone.
5. Express the Solution Using Lambert W:
By transforming and adapting our equation to fit the Lambert W function form, we find that:
[tex]\[ x + 1 = -3^x \][/tex]
Transitions into a form involving Lambert W, leading us to a solution that can be represented by:
[tex]\[ x = -1 - \frac{W(\log(3)/3)}{\log(3)} \][/tex]
Hence, the solution to the equation [tex]\( x + 5 = -3^x + 4 \)[/tex] is:
[tex]\[ x = -1 - \frac{W(\log(3)/3)}{\log(3)} \][/tex]
This expression represents the value of [tex]\(x\)[/tex] in terms of the Lambert W function, accurately capturing the solution for the given equation.
