We want to get an idea of how the solution set for the exercise's inequality behaves as c varies over the real numbers. To do so, we can graph y_1= x+c and y_2 = log x using different c values. We will analyze the functions' behavior graphically, and then we will summarize our conclusions.
As we can see, there is a critical c-value C for which y_1 is tangent to y_2. This happens because, while a linear function has a constant slope, a logarithmic function's slope is changing. Then, there is an x-value for which both have the same slope, and a c-value that translates the line to intersect the logarithmic function just once.
Furthermore, note that the logarithmic function's slope is decreasing. Then, from a certain x-value, its slope is less than the slope of the linear function. Consequently, for c < C the functions intersect twice. This is because the linear function is less than the logarithmic at the beginning, but it grows over it later.
On the other hand, if c > C there is no solution for the inequality, as the linear function will always be above the logarithmic function.
Conclusions
There is a critical c-value C, such that both functions are tangent. For c > C, the inequality x + c < log x has no solution. But, for c < C the inequality has a solution set, which increases while c decreases.
