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Form 2 Questions and Answers on Indices and Logarithms
Write in form of #a^n# where a=3
(i) 27
(ii) 9
(iii) 3
(iv) 1
(v) #1/3#
(vi)# 1/9#
(vii) #1/27#
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1.
Solve for x in: #9^x+3^(2x)-1=53#
2.
Solve the equation : #9^(x+1)+3^(2(x+1))=36#
3.
Find the value of x in the following equation: #49^(x+1)+7^(2x)=350#
4.
Given that #9^(2y) ×2^x=72#, Find the value of x and y
5.
Two numbers p and q are such that #p^3×q=189#. Find p and q
6.
Given that #9^(4/3)=3^n#,find the value of n
7.
Find the value of x which satisfies the equation: #16^(x^2)=8^(4x-3)#
8.
Find the value of m in the following equation: #(1/27)^m×(81)^-1=243#
9.
Solve for x in the equation: #(81^(2x)×27^x)-:9^x=729#
10.
Without using a calculator or mathematical tables, evaluate #(27)^(2/3)×(81/16)^(-1/4)#
11.
Simplify #(27^(2/3)-:2^4)-:32^(-3/5)#
12.
Without using logarithm tables or calculators, evaluate, #(64^(-1/2)×27000^(2/3))-:(2^-4×3^0×5^2)#
13.
Solve for x: #4^(x+1)=32#
14.
Solve for x in the equation: #32^((x-3))× 8^((x+4))=64-:2^x#
15.
Simplify: #(2^x×5^(2x)÷2^(-x))^(1/2)#
16.
Find the value of y in the equation: #(243×3^(2y))/ (729×3^y-:3^((2y-1))) = 81 #
17.
Simplify: #(243^(-2/5)×125^(2/3)) /(9^(-3/2))#
18.
Simplify: #(25^(3/4)×0.9^2×2^2)/(5^(5/2)×3^3)#
19.
Given that #6^(2(n-3))=7776#, find the value of n.
20.
Given that P=3y express the equation #3^(2y-1)+2×3^(y-1)=1# in terms of P. Hence or otherwise find the value of y in the equation #3^(2y-1)+2×3^(y-1)=1#
21.
Use logarithms to evaluate: #root(3)((35.6 times 0.0613^2))#
22.
Evaluate: #frac{1.34}{(5.24^0.8)times 0.0029}#
23.
Use mathematical tables to evaluate: #sqrt(( frac{2.935times0.0765}{32.74}))#
24.
Use logarithms to evaluate: #root(3)((frac{7.08}{76.8times7.034}))#
25.
Use mathematical tables to evaluate: #frac{sqrt 0.0645}{0.0082}#
26.
Use logarithms to evaluate #root(4)(frac{4.562 times0.038}{0.82})#
27.
Use logarithms to evaluate #frac{(0.07284)^2}{root(3)(0.06195)}#
28.
Use logarithms to evaluate #root(3)(frac{36.15times0.02573}{1.938})#
29.
Use logarithms to evaluate: #frac{(1934)^2 times sqrt0.00324}{436}#
30.
Use logarithms to evaluate: #55.9div (0.2621 times 0.01177)^(1/5)#
31.
Use logarithms to evaluate #(frac{6.79times0.3911}{log 5})^(3/4)#
32.
Use logarithms to evaluate #root(3)(frac{1.23times0.0089}{76.54})#
33.
Use logarithms to evaluate: #(3.256 times 0.0536)^(1/3)#
34.
Use logarithms to evaluate #frac{(0.0056)^(1/2)}{ 1.38 times 27.42}#
35.
Use logarithm tables to evaluate #frac{2347times0.04666}{root(3)(0.0924)}#
36.
Use logarithms to evaluate #frac{34.33}{sqrt(5.25times0.042)}#
37.
In this question, show all the steps in your calculations, giving your answer at each stage. Use logarithms, correct to 4 decimal places, to evaluate: #root(3)(frac{36.72times(0.46)^2}{185.4})#
38.
Use logarithm tables to evaluate: #(frac{0.032times14.26}{0.006})^(2/3)#
39.
Use logarithms, correct to 4 decimal places, to evaluate: #root(3)(frac{83.46times0.0054}{1.56^2})#
40.
Use logarithms, correct to 4 decimal places, to evaluate #root(3)(frac{1.794times0.038}{1.243})#
41.
Find the numerical value of: (a) #9^(1/2)# (b) #25^(-1/2)# (c) #36^(3/2)# (d) #8^(-5/3)#
42.
Find the numerical value of each of the following: (a) #(3^(-2))^3# (b) #(49^2)^(-1/2)# (c) #(10^3)^(5/3)#
43.
Simplify each of the following expressions leaving your answer in index form. (a) #3^3×3^4# (b) #5^2×5^(-3)# (c) #3^(3/2)×3^(-3/4)×3^(1/4)# (d) #6^2×7^(-4)×(8^(-2))^2×6^3×7^2×8^4#
44.
Simplify the following expressions leaving your answer in index form. (a) #512^(2/3)×8^(1/3)div2^6# (b) #(2^(-2)×(2^2)^(-6))/(2^(-4)×2^(-6) )# (c) #(6/7)^(-2)×(7/6)^(-2)#
45.
Simplify the following: (a) #r^3×r^2 t^2×r^(-3) t^(-2)# (b) #(a^n×a^m)/(a^(-n)×a^(-m) )# (c) #(x^2 y^3×x^4 y^2)/(x^3 y^4 )# (d) #(a^4 b^3×a^2 c^4×b^(-2) c^3)/(a^(-3) b^3 c^3 )#
46.
Simplify: (a) #(25y^9 k^(-7))/(81y^7 k^9 )# (b) #(3^(-4) a^4 b^6)/(3a^5 b^8 )# (c) #((a^(-n))^m×a^(-2m))/(a^mn×a^2m )#
47.
Simplify the expressions: (a) #(9a^2)^(1/3)/6^4 ×(4b^2)^(1/3)/5a# (b) #root3((27x^3 y^9)/(x^6 y^3 ))# (c) #(root5(32y^10 k^15 2^10 ) )/root4((16y^12 k^16 2^4 ) )#
48.
Solve each of the following equations: (a) #(3^2x)^3=3^4×3^8# (b) #(7^5)^x=(7^4)^xdiv7^2# (c) #(64^2ydiv16^y)/(128^y×4^2y )#
49.
Find the logarithms to base of 10 of: (a) 0.149 (b) 0.0000784 (c) 8.2 (d) 475000
50.
Find the antilogarithms of the following numbers. (a) #bar1#. 1080 (b) #bar4#. 8938
51.
Calculate: (a) #bar2# . 34 + #bar4# . 30 (b) #bar3# . 79 - #bar2# . 24
52.
Use logarithms and antilogarithm tables to evaluate: (a) 251 × 367 (b) 7.32 × 199
53.
Use logarithms and antilogarithm tables to solve: (a) 1500 ÷ 750 (b) 43.1 ÷ 3.17
54.
Use logarithms and antilogarithm tables to evaluate the following: (a) #(291 ×681)/372# (b) #(1024 ×6561)/(4096 ×729)#
55.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 478 (b) 0.0067
56.
Use tables to evaluate: (a) #sqrt((3.142 ×2.718)/(6.49 ×81.2))# (b) #(41.56 ×52.3)/sqrt42.88#
57.
Find the numerical value of: (a) #36^(-1/2)# (b) #35769^0# (c) #81^(1/2)# (d) #27^(-1/3)# (e) #16^(5/2)# (f) #100^(9/2) # (g) #64^(4/3)# (h) #256^(3/4)# (i) #729^(1/3)# (j) #(216^2 )^(1/3)# (k) #243^(2/3)# (l) #4096^(-3/4)# (m) #64^(-1/(6 ))#
58.
Simplify each of the following expressions. You may leave your answer in index form: (a)# 2^3×2^5×2^(-4) # (b)# 20^(-3)×25^2×20^3×25^(-4)# (c)#7^(-3)×8^4×7^2×8^(-3)# (d)#27^(1/3)×9^(1/2)# (e)#(3/4)^2×(3/4)^3# (f) #(20^(-2)×20^(-2)×20^(-2))/20# (g)# 2^(-6)×5^(1/2)×2^7÷2^(-4)#
59.
Simplify: #(a) x^2×x^4# #(b) y^3×y^6# #(c) n^13×n^(-7)# #(d) a^9×a^3# #(e) a^(-2)×3a^4# #(f)(2n)^3×(2n)^5# #(g) x^16÷n^12# #(h) a^15÷a^14# #(i) a^2×b^2×a^4#
60.
Simplify: (a)# b^2×x^2×b^11# (b)# b^3×n^2×b^0×n^3# (c)# a^5×c^2×a^3×b^3×a^4# (d)# a^(-2)×b^2×a^(-8)×b^(-4)# (e)# x^2 y^2× x^2 y^5× x^(-3) y^(-9)# (f)#(xyt)^n÷(xyt)^2# (g) #(a^x×a^2x)/(a^x×a^x×a^x )#
61.
Simplify: (a)#(12a^5 b^3)/(4a^3 b)# (b)#(a^(-a)×b^(-a)×c^(-a))/(a^(-2a)×b^(-2a)×c^(-2a) )# (c)#(rs^2 t^3×r^4 s^5 t^(-6)×r^7 s^8 t^(-9))/(r^10 s^11 t^(-12)×r^13 s^(-14) t^15 )# (d)#((16a^2 )^(1/2)×(36a^4 )^(-1/2))/(2a^(1/2)×5a^(3/2)×8a^(9/4) )#
62.
Solve each of the following equations: (a) #(3^(2x) )^4=81# (b) #4^(5x)÷(2^(3x ))^2=256# (c) #9^(2x)=729# (d) #2^(8x)=512# (e) #(7^4 )^(2x)=(7^4 )^3# (f) #(5^3 )^a×(5^8 )^a= 5^72#
63.
Solve each of the following equations: (a) #9^(4x)÷3^(2x)=2187# (b) #(5^(2c) )^3=(5^4 )^c×625# (c)# 2^(x/4)=8# (d)# 3^((2x-5))=27# (e)# 3^(4x)÷3^(-7)=3^15# (f) #2×3^x=162#
64.
Write in form of #a^n# where a=2 (i) 2 (ii) 4 (iii) 1 (iv)# 1/2# (v) #1/4#
65.
Write in form of #a^n# where a=3 (i) 27 (ii) 9 (iii) 3 (iv) 1 (v) #1/3# (vi)# 1/9# (vii) #1/27#
66.
Write in form of #a^n# where a=10 (i) 1000 (ii) 100 (iii) 10 (iv) 1 (v) #1/10# (vi)# 1/100# (vii) #1/1000#
67.
Copy and complete the table below:
68.
Write in logarithmic form: (a) #3^2 = 9# (b) #2^4 = 16# (c) #3^3 = 27# (d) #2^5 = 32 # (e) #3^4 = 81# (f) #5^3=125# (g) #10^0= 1# (h) #2^10 = 1024 # (i) #a^n=b#
69.
Write each of the following in index form: (a) #log_2 8 = 3# (b) #log_4 16 = 2# (c) #log_5 125 = 3# (d) #log_10 8 = x# (e) #log_b a = c# (f) #log_3 27 = 3# (g) #log_6 216= 3# (h) #log_x 40 = y# (i) #log_y x = 2# (j) #log_10 10000 = 4# (k) #log_4 6 = y# (l) #log_2 16 = 4#
70.
Use logarithm table to express each of the following numbers in the form #10^x#. (a) 681.4 (b) 1.47 (c) 4.73 (d) 7.25 (e) 9.83 (f) 5.672 (g) 8.137 (h) 3.142 (i) 2.718
71.
Use logarithm table to express each of the following numbers in the form #10^x#. (a) 3.333 (b) 12.3 (c) 59.7 (d) 82.9 (e) 72 (f) 96.1 (g) 431.5 (h) 7924 (i) 1025
72.
Use logarithm table to express each of the following numbers in the form #10^x#. (a) 1913 (b) 4937 (c) 273.7 (d) 3,910,000 (e) 958,312
73.
Find the logarithms to base 10 of: (a) 0.2843 (b) 0.3520 (c) 0.4286 (d) 0.0694 (e) 0.0485 (f) 0.0239 (g) 0.000376 (h) 0.0000523
74.
Find the antilogarithms of the following numbers: (a) #bar1 .2480# (b) #bar2 .3927# (c) #bar2 .5403# (d) #bar3 .6503#
75.
Find the antilogarithms of the following numbers: (a) #bar3 .9750# (b) #bar2 .5658# (c) #bar5 .4533# (d) #bar3 .6821#
76.
Calculate: (a) #bar2 .37+1.20# (b) #bar5 .63+bar2 .57# (c) #bar7 .12-bar5 .54# (d) 5.27 -#bar2 .73#
77.
Use logarithm and antilogarithm tables to evaluate: (a) #4192times3078# (b) #21.47times362.1#
78.
Use logarithm and antilogarithm tables to evaluate: (a) #26.1times91.2times45.7# (b) #33.2times172times44.32#
79.
Use logarithm and antilogarithm tables to evaluate: (a) #1527times3196times4157# (b) #7.312times49.45times157.2#
80.
Use logarithm and antilogarithm tables to evaluate: (a) #16.31×152.1×3290# (b) #100×245×175×396#
81.
Use logarithm and antilogarithm tables to evaluate: (a) #2145div560# (b) #412div241# (c) #882div144# (d) #3612div452#
82.
Use logarithm and antilogarithm tables to evaluate: (a) #1111div222# (b) #3.142div2.718# (c) #9250div4312# (d) #6380div2137#
83.
Use logarithm and antilogarithm tables to evaluate: (a) #(634×436×688)/784# (b) #(3041×3211)/2112#
84.
Use logarithm and antilogarithm tables to evaluate: (a) #(788×576)/675# (b) #(294+578)/(368-275)#
85.
Use logarithm and antilogarithm tables to evaluate: (a) #(839+672)/(762+393)# (b) #(3267-37)/(2236+89)#
86.
Use logarithm and antilogarithm tables to evaluate: (a) #(29^2×33^3)/64^4 # (b) #(18^2×391^4)/(15^3×56^4 )#
87.
Use logarithm and antilogarithm tables to evaluate: (a) #876.9/(61.2times3.85)# (b) #518.3/(29.5times714)# (c) #(48.35times125.3)/(39.3times50.4)#
88.
Use logarithm and antilogarithm tables to evaluate: (a) #(3.74times7.82)/5.4# (b) #399.6/(15.2times4.83times1.98)# (c) #734.4/(25.1times6.34times3.4)#
89.
Use logarithm and antilogarithm tables to evaluate: (a) #(94.7times16.45)/(12.5times8.93)# (b) #(4.48)^3/(3.16)^2 # (c) #(171.5)^3/(56.3times26.98)# (d) #(79.36)^3/(8.2times(9.2)^2 #
90.
Use logarithms to evaluate: (a) #0.9063times3.387# (b) #0.5060times0.05707# (c) #36.65times0.4163times0.007022# (d) #26.68div255.4#
91.
Use logarithms to evaluate: (a) #3.404div628# (b) #(0.0075times0.8181)/0.00509# (c) #77.9/(0.988times9100)#
92.
Use logarithms to evaluate: (a) #(2.89times5.27)/(62.25times1.908)# (b) #(34.53times361.6)/(343.7times615.8)# (c) #(91.3div18.26)/(75.4div12.09)#
93.
Divide each of the following logarithms by 2. (a) 0.8938 (b) 1.8624 (c) #bar2 .9754# (d) #bar1 .7076# (e) #bar1 .8538 # (f) #bar2 .3502# (g) #bar2 .1644# (h) #bar3 .4928# (i) #bar3 .6946# (j) #bar4 .7938#
94.
Divide each of the following logarithms by 3. (a) 0.8938 (b) 1.8624 (c) #bar2 .9754# (d) #bar1 .7076# (e) #bar1 .8538 # (f) #bar2 .3502# (g) #bar2 .1644# (h) #bar3 .4928# (i) #bar3 .6946# (j) #bar4 .7938#
95.
Divide each of the following logarithms by 4. (a) 0.8938 (b) 1.8624 (c) #bar2 .9754# (d) #bar1 .7076# (e) #bar1 .8538 # (f) #bar2 .3502# (g) #bar2 .1644# (h) #bar3 .4928# (i) #bar3 .6946# (j) #bar4 .7938#
96.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 2461 (b) 3572 (c) 4683 (d) 0.0346
97.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 0.00457 (b) 0.00072 (c) 78.039 (d) 361.472
98.
Use tables to evaluate: (a) #root2(((3.45×16.7)/31.5) )# (b) #root3(((1.794×0.038)/12.43) )# (c) #root3(((39.51×725)/0.758) )#
99.
Use tables to evaluate: (a) #root3(((4862×725)/(6437×1024)) )# (b) #root4(((6978×25.1)/132.7) )#
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