Index | Mastering Maths
Worksheet 7.1
Common Logarithms
1. Find the values of the following common logarithms without using a calculator.
(a) log 1000 (b) log 0.0001 (c) [pic]
Solution
∵ 1000 = 10( )
∴ log 1000 = ( )
2. In each of the following, find the value of y correct to 3 significant figures.
(a) [pic] (b) [pic] (c) [pic]
Solution
[pic]
3. In each of the following, find the value of x correct to 3 significant figures if necessary.
(a) [pic] (b) [pic]
Solution
[pic]
(c) [pic] (d) [pic]
Find the values of the following expressions without using a calculator. (4 ( 6)
4. (a) [pic] (b) [pic]
5. (a) [pic] (b) [pic]
6. (a) [pic] (b) [pic]
Simplify the following expressions, where x > 0 and x ( 1. (7 – 10)
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. Given that log 3 = a, express the following in terms of a.
(a) log 9 (b) [pic] (c) [pic]
Solution
[pic]
(d) [pic] (e) [pic] (f) [pic]
Additional Questions
Basic Questions
1. Complete the following table.
|(a) |Exponential form (x = 10y) |Logarithmic form (log x = y) |
| |1 = 100 | |
| |100 = 102 | |
| | |log 0.001 = (3 |
|(b) |Exponential form (x = 10y) |Logarithmic form (log x = y) |
| |5 = 10p | |
| | |log q = 1 |
| | |log r = (6 |
2. In each of the following, put a ‘(’ in the box if the given equality is valid, or else put a ‘(’.
(a) [pic] ( (b) [pic] (
(c) [pic] ( (d) [pic] (
Find the values of the following expressions without using a calculator. (3 ( 5)
3. [pic] 4. [pic] 5. [pic]
Simplify the following expressions, where x > 0 and x ( 1. (6 – 7)
6. [pic] 7. [pic]
Harder Questions
Find the values of the following expressions without using a calculator. (8 – 10)
8. [pic] 9. [pic] 10. [pic]
Simplify the following expressions, where x > 0, y > 0 and x, y ( 1. (11 – 13)
11. [pic] 12. [pic] 13. [pic]
14. Given that log 2 = x and log 7 = y, express the following in terms of x and y.
(a) log 28 (b) log 98 (c) [pic]
(d) [pic] (e) [pic] (f) log 3.5
Worksheet 7.2
Logarithms to an Arbitrary Base
1. Find the values of the following logarithms without using a calculator.
(a) [pic] (b) [pic] (c) [pic]
Solution
∵ 25 = 5( )
∴ log5 25 = ( )
2. In each of the following, find the value of x.
(a) [pic] (b) [pic] (c) [pic]
Solution
[pic]
Find the values of the following expressions without using a calculator. (3 – 6)
3. (a) [pic] (b) [pic]
4 (a) [pic] (b) [pic]
5 (a) [pic] (b) [pic]
6 (a) [pic] (b) [pic]
Simplify the following expressions, where x > 0 and x ( 1. (7 – 10)
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. Given that [pic], express the following in terms of k.
(a) [pic] (b) [pic] (c) [pic]
(d) [pic] (e) [pic] (f) [pic]
Additional Questions
Basic Questions
1. Complete the following table.
| |Exponential |Logarithmic | |Exponential |Logarithmic |
| |form |form | |form |form |
| |(x = ay) |(loga x = y) | |(x = ay) |(loga x = y) |
|(b) | |log4 4 = 1 | (e) |0.008 = 0.23 | |
|(c) |216 = 63 | | (f) |[pic] | |
2. Find the values of the following logarithms correct to 3 significant figures.
(a) [pic] (b) [pic] (c) [pic]
Find the values of the following expressions without using a calculator. (3 – 5)
3. [pic] 4. [pic] 5. [pic]
Simplify the following expressions, where x > 0 and x ( 1. (6 – 7)
6. [pic] 7. [pic]
Harder Questions
Find the values of the following expressions without using a calculator. (8 – 10)
8. [pic] 9. [pic]
10. [pic]
Simplify the following expressions, where x > 0, y > 0 and x, y ( 1. (11 – 13)
11. [pic] 12. [pic]
13. [pic]
14. Given that log 3 = x and log 5 = y, express the following in terms of x and y.
(a) [pic] (b) [pic] (c) [pic]
(d) [pic] (e) [pic]
Worksheet 7.3
Logarithmic Equations
Solve the following logarithmic equations. (1 – 6)
1. [pic] 2. [pic]
Solution
[pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
Solve the following exponential equations and give your answers correct to 3 significant figures. (7 – 10)
7. [pic] 8. [pic]
Solution
[pic]
9. [pic] 10. [pic]
Additional Questions
Basic Questions
Solve the following logarithmic equations. (1 – 6)
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
Solve the following exponential equations and give your answers correct to 3 significant figures. (7 – 8)
7. [pic] 8. [pic]
Harder Questions
Solve the following logarithmic equations. (9 – 12)
9. [pic] 10. [pic]
11. [pic] 12. [pic]
Solve the following exponential equations and give your answers correct to 3 significant figures. (13 – 16)
13. [pic] 14. [pic]
15. [pic] 16. [pic]
Worksheet 7.4
Logarithmic Functions and their Graphs
1. Refer to the graph of [pic] and answer the following.
(a) The range of values of a is .
(b) Does the graph have any x-intercept and y-intercept?
If yes, write down the intercepts.
x-intercept ( yes, ( no
y-intercept ( yes, ( no
(c) As x increases, the value of y
(increases / remains unchanged / decreases).
2. Write down the logarithmic function whose graph is symmetrical to the graph of each of the following logarithmic functions about the x-axis.
(a) [pic] (b) [pic] (c) [pic]
3. Write down the logarithmic function whose graph is symmetrical to the graph of each of the following exponential functions about the line y = x.
(a) [pic] (b) [pic]
In each of the following, with the help of the given graph of logarithmic function, sketch the required graph on the same graph paper. (4 – 7)
4. graph of [pic] 5. graph of [pic]
[pic] [pic]
6. graph of [pic] 7. graph of [pic]
[pic] [pic]
8. The figure shows the graphs of [pic], [pic] and [pic]. They intersect at the same point A.
(a) Write down the logarithmic function represented by the
graphs G1, G2 and G3 respectively.
(b) Write down the coordinates of A.
9. The figure shows the graph of [pic].
(a) Using the graph, find the value of [pic].
(b) Solve [pic] graphically.
Additional Questions
Basic Questions
1. Refer to the graph of [pic] and answer the following.
(a) The range of values of a is .
(b) Does the graph have any x-intercept and y-intercept?
If yes, write down the intercepts.
x-intercept ( yes, ( no
y-intercept ( yes, ( no
(c) As x increases, the value of y
(increases / remains unchanged / decreases).
2. (a) Complete the table for the graph of y = log4 x. (Give your answers correct to 1 decimal
place.)
|x |0.4 |0.5 |1 |
| | |[pic] |[pic] |
| | |[pic] |[pic] |
| | |[pic] |[pic] |
| |(b) |Exponential form |Logarithmic form |
| | |([pic]) |([pic]) |
| | |[pic] |[pic] |
| | |[pic] |[pic] |
| | |[pic] |[pic] |
2. (a) ( (b) (
(c) ( (d) (
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
Harder Questions
8. [pic]
9. [pic]
10. [pic]
11. [pic]
12. [pic]
13. [pic]
14. (a) [pic]
(b) [pic]
(c) [pic]
(d) [pic]
(e) [pic]
(f) [pic]
Worksheet 7.2
1. (a) ∵ [pic]
∴ [pic]
(b) ∵ [pic]
∴ [pic]
(c) ∵ [pic]
∴ [pic]
2. (a) [pic]
(b) [pic]
(c) [pic]
3. (a) [pic]
(b) [pic]
4. (a) [pic]
(b) [pic]
5. (a) [pic]
(b) [pic]
6. (a) [pic]
(b) [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
11. (a) [pic]
(b) [pic]
(c) [pic]
(d) [pic]
(e) [pic]
(f) [pic]
Basic Questions
|1. | |Exponential form |Logarithmic form |
| | |([pic]) |([pic]) |
| |(a) |[pic] |[pic] |
| |(b) |[pic] |[pic] |
| |(c) |[pic] |[pic] |
| | |Exponential form |Logarithmic form |
| | |([pic]) |([pic]) |
| |(d) |[pic] |[pic] |
| |(e) |[pic] |[pic] |
| |(f) |[pic] |[pic] |
2. (a) [pic]
(b) [pic]
(c) [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
Harder Questions
8. [pic]
9. [pic]
10. [pic]
11. [pic]
12. [pic]
13. [pic]
14. (a) [pic]
(b) [pic]
(c) [pic]
(d) [pic]
(e) [pic]
Worksheet 7.3
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
Basic Questions
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
Harder Questions
9. [pic]
10. [pic]
11. [pic]
12. [pic]
13. [pic]
14. [pic]
15. [pic]
16. [pic]
Worksheet 7.4
1. (a) The range of values of a is 0 < a < 1.
(b) x-intercept : ( yes, 1
y-intercept : ( no
(c) As x increases, the value of y decreases.
2. (a) [pic]
(b) [pic]
(c) [pic]
3. (a) [pic]
(b) [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. (a) G1 : [pic]
G2 : [pic]
G3 : [pic]
(b) (1, 0)
9. [pic]
(a) From the graph, y = 1.3 when x = 8.
∴ [pic]
(b) From the graph, x = 2.2 when y = 0.5.
∴ The solution of [pic]is x = 2.2.
Basic Questions
1. (a) The range of values of a is a > 1.
(b) x-intercept : ( yes, 1
y-intercept : ( no
(c) As x increases, the value of y increases.
2. |(a) |x |0.4 |0.5 |1 |2 |4 |7 |8 | | | |y |–0.7 |–0.5 |0.0 |0.5 |1.0 |1.4 |1.5 | |
(b)(c) [pic]
3. (a) figure a and figure b
(b) figure a and figure c
4. [pic]
(a) From the graph, y = –1.6 when x = 3.
∴ [pic]
(b) From the graph, x = 1.5 when y = –0.6.
∴ The solution of [pic]is x = 1.5.
Harder Questions
5. (a) (i) The graph of the function is the image of the graph of [pic] after reflecting along the x-axis.
(ii) The graph of the function is the image of the graph of [pic]after reflecting along the line
y = x.
(b) (i) The graph of the function is the image of the graph of [pic]after reflecting along the x-axis.
(ii) The graph of the function is the image of the graph of [pic]after reflecting along the line y = x.
6. (a) By substituting [pic] into [pic], we have
[pic]
∴ The coordinates of A are (0, 1).
(b) [pic]
(c) [pic]
7. (a) [pic]
(i) From the graph, y = 0.4 when x = 2.
∴ [pic]
(ii) From the graph, y = 0.9 when x = 5.
∴ [pic]
(b) [pic]
(i) [pic]
From the graph, x = 4.2 when y = 0.8.
∴ The solution of [pic]is x = 4.2.
(ii) [pic]
From the graph, x = 0.6 when y = –0.3.
∴ The solution of [pic]is x = 0.6.
Worksheet 7.5
1. The corresponding sound intensity level
[pic]
2. Let I W/m2 be the corresponding sound intensity.
[pic]
∴ The corresponding sound intensity is[pic].
3. Let I W/m2 be the sound intensity of the noise without the sound barrier.
Then the sound intensity is 0.2I with the sound barrier.
Reduction in the sound intensity level
[pic]
4. [pic]
∴ The energy released by the earthquake is [pic]J.
5. Let E1 and E2 be the energy released by the earthquake in city A and city B respectively. [pic]
[pic] ......(1)
[pic] ......(2)
(1) – (2): [pic]
∴ The ratio of the energy released by the earthquake in city A to that of city B is 15.8 : 1.
6. (a) When t = 10,
[pic]
∴ The population of the species 10 years after 2013 is 497.
(b) When P = 200,
[pic]
∴ The population of the species will first fall below 200 after 17 years, i.e. in 2030.
7. (a) When n = 1,
[pic]
∴ One day after the video is uploaded, the number of views of the video clip is 400.
(b) When V = 1000,
[pic]
∴ 121 days after the video is uploaded, the number of views will reach 1000.
8. (a) [pic]
∴ [pic]
(b) ∵ y-intercept of the line = 6
∴ [pic]
∵ Slope of the line[pic]
∴ [pic]
9. (a) ∵ Slope of the line[pic]
∴ [pic]
By substituting [pic] and [pic] into [pic], we have
[pic]
(b) By substituting [pic] and [pic] into [pic], we have
[pic]
Basic Questions
1. The corresponding sound intensity level
[pic]
2. (a) The corresponding sound intensity level
[pic]
(b) Let I W/m2 be the new sound intensity.
[pic]
∴ The new sound intensity is 0.574 W/m2.
3. [pic]
4. (a) In 2018, [pic] and [pic].
[pic]
(b) By substituting [pic] and [pic] into [pic], we have
[pic]
∴ The number of books in the library will reach
10 000 after 35 years, i.e. in 2048.
5. (a) [pic]
∴ [pic]
(b) ∵ y-intercept of the line = 3
∴ [pic]
∵ Slope of the line[pic]
∴ [pic]
Harder Questions
6. Let IA W/m2 and IB W/m2 be the sound intensities in the house and the construction site respectively.
[pic]
∴ The sound intensity in the construction site is 1000 times to that in the house.
7. Let E1 and E2 be the energy released in the earthquake in town A and town B respectively.
Let M be the magnitude of the earthquake in town A on the Richter scale.
[pic] ......(1)
[pic] ......(2)
By substituting [pic]into (1), we have
[pic] ......(3)
(2) – (3): [pic]
∴ The magnitude of the earthquake in town A is 7.62 on the Richter scale.
8. Let $V0 be the original value of the flat, and $V be the value of the flat after n years. Then
[pic]
When V = 1.5V0,
[pic]
∴ The value of the flat will first exceed 1.5 times its original value after 14 years.
9. (a) ∵ y-intercept of the line = 4
∴ [pic]
∵ Slope of the line[pic]
∴ [pic]
(b) By substituting [pic] and [pic] into [pic], we have
[pic]
(c) When x = 2,
[pic]
-----------------------
NF
Name: ____________________
Class: __________
If x = 10y, then y = log x.
If y = log x, then x = 10y.
log (MN) = log M + log N
[pic]
log Mn = n log M
NF
Name: ____________________
Class: __________
If y = loga x, then x = ay.
loga (MN) = loga M + loga N
[pic]
loga Mn = n loga M
[pic]
Name: ____________________
Class: __________
If loga x = y,
then x = ay.
NF
If loga x = loga y,
then x = y.
Take common logarithms on both sides of the equation.
NF
Name: ____________________
Class: __________
figure a
figure c
figure b
NF
Name: ____________________
Class: __________
log y is a linear function of x.
log y is a linear function of log x.
(b)
(a)
(b)
(a)
(b)(ii)
(b) (i)
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.