Why are logarithmic functions so useful in finance and economics?

I think there are three main reasons behind this, all related to the properties of the logarithmic function. I am not a Maths major so I will not be able to say what the relationship is (if any exists) between the three properties. 1) First of all a derivative of a log variable with respect to time will give you the growth rate or percentage change of that variable. So logarithmic variables become very handy when you are dealing with variables which change respect to time. 2) Secondly, and this is probably the most important, logarithmic functions are extremely convenient for Taylor Approximation. The reason behind this is I think the logarithmic curve is very smooth and almost coincidental with the slope for any segment on the curve. As a result first order Taylor polynomial is enough to approximate a log function. This property comes in very handy in log linearization. First of all this property helps in finding out speed of convergence in growth theories i.e. how fast an economy which has fallen out of a steady state  will return to the steady state. Secondly, it helps in comparing two sets of mean reverting variables. How? Well if you have two mean reverting variables and there is reason to believe that the variance of the two variables are related even though at the levels the variables differ greatly from one another in value then you can get rid off that problem by log linearizing. Think of two variables which are related such that a shock in one variable will also cause a shock in the other variable but the shocks are temporary and over a long period of time both the variables will be mean reverting and at their levels there is no relationship between the variables. So you will want to find a way so that you are just left with the shocks and get rid of the means of the two variables. Log linearization gives you that opportunity. 3) A third point is that logarithms help in scaling down values of variables. This is more from the point of view of data work. If your data has very large deviations then taking logs instead of the base value helps in making charts and graphs.

The main reason is that historical data shows that many phenomena in economics (especially microeconomics) and finance follow what my professors called "the well known S-shaped curve". This is represented by logarithmic functions.

Finance and economics are directly related to inflation ( increase in price ). Increase in price behaves just like governed by formula. It increases by certain value after every year. This is for economics. Similarly for finance whenever you invest in conventional investment policies like fixed deposits, provident funds ,investments in valuable materials their price increases just as rate of inflation . Inflation rate is generally the value that can be expressed in more better manner by using exponential function or logarithmic function.