How do I go about finding the regulator voltage with only the information I've been given?

I tried writing a KCL and expressing everything in terms of Vd but my diode current contains an expontential term which is going to make it hard to isolate and solve Vd.

Is there an easier way? There is no solution given for this question which makes me think something is fishy.

Any ideas?

 

Look at the typical forward voltage curves in the datasheet of a 1N4148 diode.
There are 3 curves that go from a current of only 1uA to a current as high as 800mA.
Up to about 100mA the curves are a straight line when the current increases logarithmically.
There is another curve that shows how the forward voltage is changed by temperature changes.

In your circuit the load changes the current in the diodes. Reduced current in the diodes also might cause the temperature of the diodes to decrease which increases their forward voltage a little.