Hi there,
This popped up in my exam the other day, absolutely no idea how to solve it using secondary school knowledge:
2^x = -x
The equation You have written is trascendental and its solution has to be found numerically. A non well popular but elegant way to solve it is to writre the equation as $\displaystyle f(x)= -2^{x}-x=0$ [th sign '-' has a precise scope...] and then compute for n large enough the n-th term of the sequence defined by the 'recursive relation' ...
$\displaystyle x_{n+1}=-2^{x_{n}}\ ;\ x_{0}=0$ (1)
In few step You arrive at the solution $\displaystyle x \sim -.641186$. The reason for which $\displaystyle x_{n}$ tends to the [real] solution of Your equation will be explained in a tutorial post written in the section 'Discrete mathematics'...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
The equation...
$\displaystyle f(x)=- 2^{x}-x=0$ (1)
... cannot be solved using simple log laws!... once You have found in graphical way that the solution is between -1 and 0, if You need better accuracy the only choice You have are numerical methods. Among them one of the most simple [and elegant...] even if not widely used is to consider the solution as the limit [if it exists...] of the sequence defined by the recursive relation...
$\displaystyle \Delta_{n}= x_{n+1}-x_{n}= f(x_{n})$ (1)
In Your case the (1) is...
$\displaystyle x_{n+1}= -2^{x_{n}}$ (2)
What You have to do now is to compute iteratively the $\displaystyle x_{n}$ till to an n 'large enough'. Starting with $\displaystyle x_{0}=0$ You easily find...
$\displaystyle x_{1}= -2^{0}= -1$
$\displaystyle x_{2}= -2^{-1}= -.5$
$\displaystyle x_{3}= -2^{-.5}= -.707106...$
$\displaystyle x_{4}= -2^{-.707106...}= - .612547...$
$\displaystyle x_{5}= -2^{-.612547...}= -.65404...$
$\displaystyle x_{6}= -2^{-.65404...}= -.6354978...$
$\displaystyle x_{7}= -2^{-.6354978...}= -.6437186...$
$\displaystyle x_{8}= -2^{-.6437186...}= -64006102...$
... and so one. The convergence to the solution $\displaystyle x \sim -.641186$ is evident...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Clearly you are not expected to solve it by hand, for the simple fact that it cannot be solved exactly by hand or otherwise (unless you use a special function called the Lambert W-function).
Did you have a graphics or CAS calculator in the exam? You are expected to know how to use your calculator to solve it.