Quantum computation and error correction requires a large yet controllable Hilbert space. While many implementations use discrete quantum variables such as the energy of a two-level system to encode quantum information, continuous variables could allow access to a larger computational space while minimizing the amount of required hardware. With a toolset of conditional qubit-photon logic, we encode quantum information into the amplitude and phase of coherent state superpositions in a resonator, also known as Schrödinger cat states. We achieve this using a superconducting transmon qubit with a strong off-resonant coupling to a waveguide cavity. This dispersive interaction is much greater than decoherence rates and higher-order nonlinearites and therefore allows for simultaneous control of over one hundred photons. Furthermore, we combine this experiment with fast, high-fidelity qubit state readout to perform composite qubit-cavity state tomography and detect entanglement between a physical qubit and a cat-state encoded qubit. These results have promising applications for redundant encoding in a cavity state and ultimately quantum error correction with superconducting circuits.