follow all the directions. Show work for everything. All questions must be answered in full. follow attachment. Please do in a document format and upload.

1) Use the Laws of Logarithms to solve the logarithmic equation for !.

log ! (! + 3) + log ! (! + 2) = 1

2) Write each logarithm as an exponential equation. Then, give the value of ! that satisfies the

equation.

a. log 0.01 = !

b. log ! ! = 4

c. log ” 243 = 5

3) Solve each exponential equation for !. Write the exact solution for each.

a. 32″ = 16#$”

b. 1 “$# − 5 = 5

4) a. State the transformations from the graph of 3(!) = 2” to 4(!) = 2″$% + 1.

b. Match the function, 4(!) = 2″$% + 1 with the correct graph.

(a)

(c)

(b)

(d)

5) If $12,500 is invested at an interest rate of 4.2% per year, compounded quarterly, find the value of

the investment after 20 years.

6) How long will it take $4,000 to triple if it is invested at 5.5% compounded continuously?

7) If the initial size of a population is 5& and the doubling time is 6, then the size of the population at

time 7 is

5(7) = 5′ 2(/*

where 6 and 7 are measured in the same units (minutes).

A certain culture of the bacterium Staphylococcus aureus initially has 85 bacteria and is observed to

double every 42 minutes.

(a) Find an exponential model for the number of bacteria present in the culture after 7 minutes.

(b) Find the number of bacteria after 5 hours.

(c) After how many minutes will the bacteria count reach a population of 8,500?

8. Given below is a graph of a function. Draw its inverse first. Hint: What are some points on 3(!)?

(a) What is the equation of the given function?

3(!) =

(b) What is the domain and range of 3?

Domain:

Range:

(c) What is the equation of the vertical asymptote of 3?

(d) What is the equation of 3 $# ?

3 $# (!) =

(e) What is the domain and range of 3 $# ?

Domain:

Range:

(f) What is the equation of the horizontal asymptote of 3 $# ?