LM3914 has open collector output on its pins... When a particular output is active the internal chip transistors will pull the output pin towards ground. But when the output is not active then it remains open and floating.
Thus, when connecting the LM3914's output to the input of other ICs which expecting logical high or low signals, a pull up resistors might be a good idea to avoid random states. Use a cheap 10x resistor network of 10Kilo ohm to pull up all outputs of LM3914 when connecting it to logic gates.
If active high outputs are required from ULN2003, inverters are necessary between LM3914 and ULN2003. Use either hex CMOS inverters or built simple pnp inverter for every channels. If pnps are used, connect the output of LM3914 to the transistor base, link the emitter to Vcc through 10K resistor. Connect the collector of the PNP to ULN2003. In this case, separate pull up resistors will not be required.