MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document How do you solve this equation: 10 = 2 x + xIs it possible to solve this equation?

Expert Answer to "MathJax(?): Can't find handler for document MathJax(?): Can't..."

Jonathan Miles

Answered question

2022-07-04

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document How do you solve this equation:

10 = 2^{x} + x

Is it possible to solve this equation?

a & = b^{x} + x a - x & = b^{x} \log_{b} (a - x) & = x

If a and b are known, how do you find x?

Answer & Explanation

Jenna Farmer

Beginner2022-07-05Added 17 answers

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document As noted in the comments, since the function is monotonically increasing, there is only one real solution. To find the solution for say

10 - 2^{x} - x = 0

, you can use Newton's method to iterate:

n + 1 = x_{n} - \frac{10 - 2^{x} - x}{2^{x} \ln 2 - 1}

Alternatively, you could use the Lambert W function to write the solution as:

(2^{10} \ln (2)) \ln (2)

To get to this, use the definition:

(x) \exp (W (x))

And define

y = 10 - x

, and use W(y).

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