G1 Logarithmic Functions Q and A

G1 Logarithmic Functions Q and A

1. Evaluate each of the following logarithms, if possible.

(a) (b) (c) (d)

(e) (f) (g) (h)

2. Use your calculator, together with your change of base formula where necessary, to find each of the following logarithms rounded to 5 decimal places.

(a) (b) (c) (d) (e)

(f) (g)

3. Write each of the following as the logarithm of a single simplified expression. Your answer must be in the form , where  is the simplified expression.

(a) (b)

(c) (d)

(e) (f)

4. Expand each of the following logarithmic expressions in terms of , , and .

(a) (b) (c) (d)

5. Evaluate each of the following logarithms, if possible.

(a) (b) (c) (d) (e)

(f) (g) (h)

6. Use your calculator to find the value of each of the following rounded to 5 decimal places.

(a) (b) (c) (d)

(e) (f)

7. Write each of the following as the natural logarithm of a single simplified expression. Your answer must be in the form , where  is the simplified expression.

(a) (b) (c)

(d) (e)

8. Expand each of the following logarithmic expressions in terms of , , and .

(a) (b) (c) (d)

9. Find the derivative of each of the following functions.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

10. Find the slope of the tangent line to the given function at the given value of x. Round your answer to five decimal places.

(a) at .

(b) at .

11. Find the equation of the tangent line to the graph of drawn at the point where . Leave in unsimplified form in your answer.

12. Find the derivative of each of the following functions.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

p.

q.

r.

s.

13. Use implicit differentiation to find if .

14. Suppose that the number of units of a commodity that are sold after spending x hundred dollars on advertising is given by the function .

(a) How many units are sold if $100 is spent on advertising? Round to the nearest integer.

(b) How many units are sold if $1000 is spent on advertising? Round to the nearest integer.

(c) Find .

(d) Find and interpret the result.

(e) Find and interpret the result.

(f) Explain why ?

(g) Does have any local extrema? Explain.

15. A man’s bank account after x months shows a balance given by the function .

(a) How much money was in his account initially?

(b) How much money was in his account after 10 months?

(c) Find .

(d) Find and interpret the result.

(e) How many months did it take for the account to go down to $0?

16. A salesperson at a car dealership began work in an unfamiliar city. The number of people whose names the salesperson could remember after working x weeks is given by the function .

(a) How many people did the salesperson know initially?

(b) How many people did the salesperson know after 4 weeks? Round to the nearest integer.

(c) Find .

(d) Find and . Interpret your results.

(e) As x increases, does increase or decrease? What might be the reason?

17. For each of the following functions, determine the equation of the tangent line drawn at the point .

a.

b.

c.

d.

e.

18. For each of the following functions, determine the equation of the tangent line drawn at the point . Leave e in your answer.

a.

b.

c.

19. For each of the following functions, determine the critical numbers and the local extrema.

a.

b.

c.

d.

e.

f.

20. Find the intervals in which the function is concave up and concave down.

a.

b.

G1 Logarithmic Functions Answer Key

1 (a) 3 (b) 0 (c) not possible (d) not possible

(e) 4 (f) -1 (g) -9 (h) 3/2

2(a) 1.7160 (b) -0.0055 (c) (log 4)/(log 3) = 1.2619

(d) 0.7925 (e) 2.0206(f) -2.5850 (g) 3.1439

3(a) (b)

(c) (d)

(e) (f)

4. (a) (b)

(c) (d)

5(a) 1 (b) 3 (c) 10 (d) -4

(e) 1/2 (f) 0 (g) no solution(h) no solution

6(a) 1.6094 (b) 4.6052 (c) -0.9555 (d) 1.9251

(e) 3.4342 (f) 1.5001

7(a) (b) (c)

(d) (e)

8. Expand each of the following logarithmic expressions in terms of , , and .

(a) (b) (c) (d)

9.(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

10. (a) /
(b) /
11. /

12. Find the derivative of each of the following functions.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

m.

n.

o.

p.

q.

r.

s.

13. /

14(a)

(b)

(c).

(d) If $600 is spent on advertising, the rate of increase of sales will be 33 apartments per $100 spent.

(e) If $6000 is spent on advertising, the rate of increase of sales will be 3 apartments per $100 spent.

(f) The rate that the apartment sales increases is lower with the increased advertising budget. (The law of diminishing returns)

(g) . The derivative cannot equal zero.

15(a)

(b)

(c) .

(d) The account balance is dropping at a rate of $100 per month.

(e) /

To solve for x, gives you a ridiculously large number, therefore, there is no realistic answer.

16. A salesperson at a car dealership began work in an unfamiliar city. The number of people whose names the salesperson could remember after working x weeks is given by the function .

(a)

(b)

(c).

(d)

The rate at which the salesperson is meeting new people is reducing as the weeks go by. (e) The rate decreases as there are fewer new people that the salesman would meet and be able to recall.

17. For each of the following functions, determine the equation of the tangent line drawn at the point .

a. /
b. /
c. /
d. /
e. /

18. For each of the following functions, determine the equation of the tangent line drawn at the point . Leave e in your answer.

a. /
b. /
c. /
19a. /
b. /
c. /
d. /
e. /
f. /

20a.

b.