06 Jul 1. What is a natural logarithm? How is it different from and si

1. What is a natural logarithm? How is it different from and similar to regular logarithms? Provide examples for how natural logarithms appear in nature or in natural science. What are two applications of logarithmic and exponential functions in science?2. What is the relationship between exponential and logarithmic functions? Include examples.3. Graph two logarithmic functions with different bases and their corresponding exponential functions. What are the similarities and differences in the graphs?4. WK3 Exercise 8 – Exponential FunctionsGraph the function.  Graph the function. Find the accumulated value of an investment of $8500 if it is invested for 3 years at an interest rate of 4.25% and the money is compounded monthly.Find the accumulated value of an investment of $1200 if it is invested for 6 years at an interest rate of 6% and the money is compounded continuously.MTH/225 WK3 Exercise 10 – Exponential FunctionsLet  R be the response time of some computer system, U be the machine utilization (CPU), S be the service time per transaction,Q be the queue time (or wait time… pronounced as my last name, Kieu) and  a be the arrival rate (number of log-on users). The total response time (excluding network delay) is the sum of queue time and service time. Thus,R = S + Q (1)Generally, the service time is predictable and relatively invariant. The time a transaction spends in queue, however, varies with the transaction arrival rate a. Assuming that the arrival and service processes are homogeneous (time-invariant), the following is true:R = SQ + S   (2)According to Queuing Theory (Allen, 2014): Q = a * R  (3)Manipulating equations (2) and (3) using Factoring method, we obtain:    R = S / 1-aS  (4)1. Show how you manipulate the two equations (2) and (3) to arrive at (4). 2. Create a graph for (4), discuss observations, and make interpretations of this graph.