Graphs

1. (5 marks) Evaluate each of following:

a) log2 4

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b) log3 27

c) log3 √

3

d) log5 1

e) log2

( 1

2

) 2. (10 marks) Sketch the graph of each function using transformations. State the domain and range.

a) f(x) = 3 log10(x) + 3

b) f(x) = − log10(x− 6)

c) f(x) = log10(2x)

d) f(x) = 4 log10

( 1

4 x

) − 2

e) f(x) = log10(−2x− 4)

1

 

 

3. (5 marks) Sketch the graph of

f(x) = −1

log2(x + 2)

4. (10 marks) Half-life is the time it takes for half of a sample of a radioactive element

to decay. The function M(t) = p

( 1

2

) t b

can be used to calculate the mass remaining if

the half-time is b and the initial mass is p. The half-life of radium is 1620 years.

a) If a laboratory has 5 g of radium, how much will there be in 150 years?

b) How many years will it take until the laboratory has only 4 g of radium?

5. (10 marks) Write 1

2 loga(x) +

1

2 loga(y) −

3

4 loga(z) as a single logarithm. Assume

that all the variables represent positive numbers.

6. (10 marks) Solve for x.

a) 4x+1 + 4x = 160

b) 2x+2 + 2x = 320

c) 10x+1 − 10x = 9000

d) 3x+2 + 3x = 30

7. (10 marks) Solve following equations.

a) loga(x + 2) + loga(x− 1) = loga(8 − 2x)

b) log(35 −x3) log(5 −x)

= 3

c) log5(log3(x)) = 0

d) log2(log4(x)) = 1

8. (10 marks) The intensity, I, of light passing through water can be modelled by the equation I = 101−0.13x, where x is the depth of the water in metres. Most aquatic plants require a light intensity of 4.2 units for strong growth. Determine the depth of water at which most aquatic plants receive the required light.

9. (10 marks) How is the instantaneous rate of change affected by the changes in the parameters of the function?

a) y = a log[k(x−d)] + c

b) y = ab[k(x−d)] + c

10. (10 marks) Evaluate following expressions.

a) log(15) + log(40) − log(6)

2

 

 

b) log7(343) + 2 log7(49)

11. (10 marks) The equation that models the amount of time, t, in minutes that a cup of hot chocolate has been cooling as a function of its temperature, T , in degree Celsius

is t = log

( T − 22

75

) ÷ log(0.75). Calculate the following.

a) the cooling time if the temperature is 35◦C

b) the initial temperature of the drink

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