Problem: Logarithm of an exponential function

Problem

A variable \(y\) is equal to \(c\) at \(t=0\) and it grows by \(a%\) per time period. Thus, at time \(t\) , \(y=c{\left( 1+ \frac{a}{100} \right)}^t\) . Show that \(log \left( y \right)\) is a linear function with intercept \(log \left( c \right)\) and a slope which is approximately \(a/100\) .

This is the main reason many economic variables are logged. Any variable that tend to grow exponentially, such as GDP, prices and so is easier to understand logged. It becomes close to linear and the slope is a measure of the growth rate.