Here is a hint
Where would your representation of negative numbers connect with the representation of positive numbers? If you want/need a smooth transition between the two, they should connect at one point (0 according to your specification). This means that there must be a specific value of x at which you can pass from the blue curve to the red curve in a continuous fashion. Yet, if you look at the graph of what would be a trivial linear scale and a log base 10 scale, there is no such point.
Now you can try to find a transformation which which connects, but your scale wont be as simple as a linear increments of exponents of 10. Yet you can compute the location of exponents of 10, and place ticks and appropriate labels at these positions. But it is not direct and you would have to work a bit for that.
