Mhf4u Exponential And Logarithmic Test 1e2a4u

MHF4U, December 2017

Time Completed______

Name_________________

Unit Test 5 – Exponential and Logarithmic Functions K/U /20

(25%)

TIPS /24

(25%)

APP /15

(30%)

COM /10

(20%)

Part One – Knowledge and Understanding_____________________________________________ Multiple Choice – Identify the choice that best answers the following questions. 1. _____ Another way of writing 𝑥 = 𝑏 𝑦 is a) log 𝑦 𝑥 = 𝑏

b) log 𝑥 𝑏 = 𝑦

c) log 𝑏 𝑦 = 𝑥

d) log 𝑏 𝑥 = 𝑦

1 𝑥

2. _____ The function 𝑓(𝑥) = (5) has y-intercept of a) 5

b) -5

c) 1

d) -1

c) 1

d) e

3. _____ Evaluate log 0.1 + log 𝑒 𝑒 a) -1

b) 0

4. _____ Solve the equation: log 5𝑥 = 2 log 6 − log 4 a) 1.8

b) 9

c) 1.33

d) 0.6

5. _____ The equation of the vertical asymptote for the function 𝑦 = −2 log 3 (3𝑥 + 3) + 4 is a) 𝑥 = −1

b) 𝑥 = −2

c) 𝑥 = 3

d) 𝑥 = 4

6. _____ log 5 45 can be written as: a)

log 5

b) log 5 9

log 45

c)

log 9 log 1

d)

log 45 log 5

7. _____ The x-intercept of 𝑦 = 6𝑥 is a) 1

b) 6

c) 0

d) Does Not Exist

8. Evaluate without using a calculator. Show all steps for full marks. a) log 𝜋 𝜋 6

b) 7log7 25

c) 2 log 2 + log 25

/1

/1 /2

d) log 7 441 - 2 log 7 3

9. Match each of the following graphs with its function.

______ log 2 𝑥

______ log 1 𝑥 2

______ 2𝑥

1 𝑥

______ (2)

10. Expand and simplify the following expression

𝑦 = log 6

/2

(𝑍 5 )(𝑉 2 ) 4

√𝑃

.

/3

Part Two – Thinking/Inquiry/Problem Solving_________________________________________

11. Solve each of the following equations for 𝑥. Express answers as exact solutions where possible, otherwise round approximate answers to two decimal places. Also, state any extraneous roots.

a) 32𝑥+2 − 2 ∙ 3𝑥+3 + 45 = 0

/4

b) 56𝑥−3 = 3−9𝑥

/4

c) log(5 − 4𝑥) − 2 log(𝑥) = 0

/4

d) log 6 (7 − 𝑥) + log 6 (𝑥 + 6) = 2

/4

16

12. Sketch the graph of 𝑦 = log 2 and state its properties. Round your answers to two (𝑥−3)3 decimal places where appropriate. Show all steps. /8

Domain

Range

Asymptote

x-intercept

y-intercept

Part Three – Application____________________________________________________________ 13. A massive earthquake measuring 7.8 on the Richter scale that hit Mexico this past summer was the strongest in 20 years. It resulted many fatalities and billions of dollars in damage. Another earthquake hit Iraq two months later measuring 6.8 on the Richter scale. How much more intense was the earthquake that hit Mexico than the one that hit Iraq? /2 𝐼2 𝑀2 − 𝑀1 = 𝑙𝑜𝑔 ( ) 𝐼1

14. A well-dressed math teacher was inspired to work on his dance steps for the Winter Formal. He found out that, on average, the number, 𝑁, of dance moves he can learn after t days of practice is given by 𝑁 = 48(1 − 𝑒 −0.08𝑡 ). How many days of practice does it take for this mystery math teacher to learn 30 moves? /3

15. The population of OTHS alumni that attended various reunions up until 2007 was 9000. Another update in 2017 showed that this number had grown to 11500. How many alumni are expected to be attending OTHS reunions by the year 2030? /5

16. The OTHS science club has identified that the function 𝑆(𝑑) = 86 log(𝑑) + 112 relates the speed of the wind, 𝑆, in miles per hour, near the center of a tornado to the distance the tornado travels, d, in miles. a) What is the speed of the tornado after 2 miles of travel?

/2

b) What distance does the tornado need to travel to reach speeds of 200 miles per hour?

/3

Part Three – Communication_________________________________________________________

Question

Answer

A logarithm is an exponent.

T

F

The x-intercept for all logarithmic functions is 𝑥 = 1.

T

F

In the equation log 𝑏 𝑥 = 𝑦, the value of b can equal 1.

T

F

The equation log 1 4 = −2 is meaningless.

T

F

The equation log 𝑏 𝑥 = 𝑦 is not defined for 𝑥 = 1.

T

F

The equation 𝑦 = log 2 4𝑥 can be drawn using a vertical translation without applying a horizontal compression.

T

F

2

Communication Uses mathematical language, symbols, visuals, and conventions correctly.

1 mark

2 marks

3 marks

4 marks

Rarely

Sometimes

Usually

Always