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I understand that k alone is the equilibrium constant and know how to find this number or when given it I can take the log of this number, but what is the purpose of this number? What does the log of k tell us in relation to the reaction? Why do we find log k?

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$\log K$ relates to the thermodynamic changes in entropy and enthalpy of a reaction. $$\Delta H - T \Delta S = \Delta G = \Delta G^\circ - RT \ln Q$$

At equilibrium $Q = K$ and:

$$\Delta G = 0 = \Delta G^\circ - RT \ln K \Longrightarrow \Delta G^\circ = RT\ln K$$

$$\Delta G^\circ = \Delta H^\circ -T\Delta S^\circ = RT \ln K$$

and:

$$\ln K = \frac{\Delta H^\circ -T\Delta S^\circ}{RT} = \frac{\Delta H^\circ}{RT} - \frac{\Delta S^\circ}{R}$$

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The logarithm of k is equally effective to understand the thermodynamics of a reaction.As the logarithmic function is increasing, you can compare log k of two reactions and arrive at similar qualitative interpretations as you would for k.The most important thing is that the the standard free energy change for a reaction is directly related to the logarithm of k. $\Delta G^\circ = -RT \ln K$

Which actually tells you that the logarithm of k is often more useful for thermodynamic computations.

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