MHF 4U – UNIT 8 TEST – Exponential and Logarithmic Functions
NAME:
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DATE:
/ 50
Show all work (the answer is only worth half a mark) and give exact answers unless stated
otherwise.
1.
[2]
a) Write 4− 3 =
1
in logarithmic form.
64
________________
b) Write log 3 243 = 5 in exponential form.
________________
2. Evaluate without using the log key on your calculator. Show your work.
Hint: use the laws of logarithms.
1
a) 𝑙𝑜𝑔3 531441
b) 𝑙𝑜𝑔6 279936
c) 5𝑙𝑜𝑔5 12
d) 𝑙𝑜𝑔4 √1024
[4]
3.
7
Use the laws of logarithms to simplify, then evaluate. Show your work.
a) log 500 − log 5
b)
2 log 2 + 2 log 5
c) log 3 27 + log 9 3
[2]
[2]
[2]
4. Solve for x. Show your work. No trial and error. Round to 1 decimal places if
necessary.
a) 253𝑥 = 52𝑥−1
[3]
[3]
b) log 7 (𝑥 − 2) + log 7 (𝑥 + 4) = 1
MHF 4U – UNIT 8 TEST – Exponential and Logarithmic Functions
c) Solve 3𝑥 + 6(3−𝑥 ) = 5.
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d) 5
3𝑥−2
= 4𝑥+1
[3]
[4]
5. Write the following as a single logarithm.
1`
a) 2 log a + log(3b) − log c
2
[2]
6. Let f ( x) = log x and g ( x) = 7 log(8x − 32) − 4 . The point (-6, -15 ) is on f (x) ; determine
the coordinates of the corresponding transformed point on g (x) . Show your work.
[3]
7. In conducting a science experiment about the growth rate of a certain strain of bacteria, Juan
noticed that at 10:00 a.m. there were approximately 300 bacteria and at 2:00 p.m. there were
approximately 850 bacteria. Assuming that the bacteria continue to grow at the same
exponential rate, determine when there will be 3000 bacteria. Give your answer as a time, to
the nearest minute, in the form xx:xx am/pm. [3]
MHF 4U – UNIT 8 TEST – Exponential and Logarithmic Functions
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8. Determine the hydrogen ion concentration in moles per litre, of a solution with a pH of 3.2
[1]
pH = – log[H+]
9. When you drink a cup of coffee or a glass of cola, or when you eat a chocolate bar, the
percent, P, of caffeine remaining in your bloodstream is related to the elapsed time, t, in
 log P 
.
hours, by t = 5
 log 0.5 
a) How long will it take for the amount of caffeine to drop to 10% of the amount consumed? [2]
Give answer to one decimal place.
b) Suppose you drink a cup of coffee at 6:00 a.m. What percent of the caffeine will remain
in your body at noon? Give answer to the nearest whole number.
[3]
10. A savings bond offers interest at a rate of 5.2%, compounded weekly. Suppose that a $1000
bond is purchased. How long will it take for the investment to triple in value? Give answer
to one decimal place.
[3]
MHF 4U – UNIT 8 TEST – Exponential and Logarithmic Functions
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11. The parent function 𝑓(𝑥) = log 𝑥 is transformed into 𝑔(𝑥) by the following transformations:
• horizontal compression by a factor of 0.27
• reflection in the y-axis
• vertical translation up 5 units
• horizontal translation left 2 units
Write the equation of the transformed function [2]
𝒙𝟐 √ 𝒙
12. If 𝐥𝐨𝐠 𝒃 𝒙 = 𝟎. 𝟓, find the value of 𝐥𝐨𝐠 𝒃 ( 𝒃 )
1 4𝑥−2𝑦
13. If log (5𝑥−2𝑦) 65536 = 4 and (9)
and 𝑦.
[3]
[3]
= 59049, determine, algebraically, the values of 𝑥