Jamb Mathematics - Lesson Notes on Indices, Logarithms and surds for UTME Candidate

Indices, Logarithms and surds | Jamb Mathematics

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This is a reminder to dedicate time to thorough preparation for the upcoming exam. The assessment will test your understanding of key concepts and their applications, emphasizing critical thinking and problem-solving skills. Please ensure you review all relevant materials, practice problem sets, and clarify any uncertainties prior to the exam date.

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Indices, logarithms and Surd? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Number Base together and move one step closer to achieving your exam success! Blissful learning.

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Calculation problems on the laws of indices

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Question 1

Simplify:

2^3 \times 2^4

Solution:

Law: $a^m \times a^n = a^{m+n}$ .
$2^3 \times 2^4 = 2^{3+4} = 2^7$ .
$2^7 = 128$ .

Answer:

128

Question 2

Simplify:

5^6 \div 5^2

Solution:

Law: $a^m \div a^n = a^{m-n}$ .
$5^6 \div 5^2 = 5^{6-2} = 5^4$ .
$5^4 = 625$

Answer:

625

Question 3

Simplify:

(3^2)^4

Solution:

Law: $(a^m)^n = a^{m \cdot n}$ .
$(3^2)^4 = 3^{2 \cdot 4} = 3^8$ .
$3^8 = 6561$ .

Answer:

6561

Question 4

Simplify:

7^0

Solution:

Law: $a^0 = 1$ , for any $a \neq 0$ .
$7^0 = 1$ .

Answer:

1

Question 5

Simplify:

10^{-3}

Solution:

Law: $a^{-m} = \frac{1}{a^m}$ .
$10^{-3} = \frac{1}{10^3} = \frac{1}{1000}$ .

Answer:

\frac{1}{1000}

Question 6

Simplify:

4^{-2}

Solution:

$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ .

Answer:

\frac{1}{16}

Question 7

Simplify:

(2^3 \cdot 3^2)^2

Solution:

Law: $(a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}$ .
$(2^3 \cdot 3^2)^2 = 2^{3 \cdot 2} \cdot 3^{2 \cdot 2} = 2^6 \cdot 3^4$ .
$2^6 = 64$ , $3^4 = 81$ .
$64 \cdot 81 = 5184$ .

Answer:

5184

Question 8

Simplify:

\frac{3^5}{3^3}

Solution:

Law: $a^m \div a^n = a^{m-n}$ .
$\frac{3^5}{3^3} = 3^{5-3} = 3^2 = 9$ .

Answer:

9

Question 9

Simplify:

(2 \cdot 5)^3

Solution:

Law: $(a \cdot b)^m = a^m \cdot b^m$ .
$(2 \cdot 5)^3 = 2^3 \cdot 5^3$ .
$2^3 = 8$ , $5^3 = 125$ .
$8 \cdot 125 = 1000$ .

Answer:

1000

Question 10

Simplify:

\frac{4^3 \cdot 2^2}{2^3}

Solution:

Simplify numerator: $4^3 = (2^2)^3 = 2^6$ , so $2^6 \cdot 2^2 = 2^{6+2} = 2^8$ .
Divide: $\frac{2^8}{2^3} = 2^{8-3} = 2^5 = 32$ .

Answer:

32

Question 11

Simplify:

(x^3)^4

Solution:

Law: $(a^m)^n = a^{m \cdot n}$ .
$(x^3)^4 = x^{3 \cdot 4} = x^{12}$ .

Answer:

x^{12}

Question 12

Simplify:

\frac{(2^3)^2}{4^2}

Solution:

Numerator: $(2^3)^2 = 2^{3 \cdot 2} = 2^6$ .
Denominator: $4^2 = (2^2)^2 = 2^4$ .
$\frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4$ .

Answer:

4

Question 13

Simplify:

x^5 \cdot x^{-3}

Solution:

Law: $a^m \cdot a^n = a^{m+n}$ .
$x^5 \cdot x^{-3} = x^{5-3} = x^2$ .

Answer:

x^2

Question 14

Simplify:

\frac{y^{-3}}{y^{-5}}

Solution:

Law: $a^m \div a^n = a^{m-n}$ .
$\frac{y^{-3}}{y^{-5}} = y^{-3 - (-5)} = y^{-3 + 5} = y^2$ .

Answer:

y^2

Question 15

Simplify:

\frac{1}{x^{-2}}

Solution:

Law: $a^{-m} = \frac{1}{a^m}$ .
$\frac{1}{x^{-2}} = x^2$ .

Answer:

x^2

Question 16

Simplify:

(3x^2)^3

Solution:

$(3x^2)^3 = 3^3 \cdot (x^2)^3$ .
$3^3 = 27$ , $(x^2)^3 = x^{2 \cdot 3} = x^6$ .
$27x^6$ .

Answer:

27x^6

Question 17

Simplify:

\frac{x^3 \cdot y^4}{x^2 \cdot y^2}

Solution:

Simplify $x^3 \div x^2 = x^{3-2} = x$ .
Simplify $y^4 \div y^2 = y^{4-2} = y^2$ .
Result: $x \cdot y^2 = xy^2$ .

Answer:

xy^2

Question 18

Simplify:

2^5 \div (2^3)^2

Solution:

Denominator: $(2^3)^2 = 2^{3 \cdot 2} = 2^6$ .
$2^5 \div 2^6 = 2^{5-6} = 2^{-1} = \frac{1}{2}$ .

Answer:

\frac{1}{2}

Question 19

Simplify:

\frac{(x^3)^4}{x^8}

Solution:

Numerator: $(x^3)^4 = x^{3 \cdot 4} = x^{12}$ .
Divide: $\frac{x^{12}}{x^8} = x^{12-8} = x^4$ .

Answer:

x^4

Question 20

Simplify:

\frac{(a^2 b^3)^2}{b^4 a^3}

Solution:

Numerator: $(a^2 b^3)^2 = a^{2 \cdot 2} b^{3 \cdot 2} = a^4 b^6$ .
Divide $a^4 \div a^3 = a^{4-3} = a$ .
Divide $b^6 \div b^4 = b^{6-4} = b^2$ .
Result: $a b^2$ .

Answer:

a b^2

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Calculation problems on equations involving indices

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Question 1

Solve for

x

2^x = 16

Solution:

Rewrite $16$ as $2^4$ : $2^x = 2^4$ .
Equate the powers: $x = 4$ .

Answer:

x = 4

Question 2

Solve for

x

5^{x+1} = 125

Solution:

Rewrite $125$ as $5^3$ : $5^{x+1} = 5^3$ .
Equate the powers: $x + 1 = 3$ .
Solve: $x = 2$ .

Answer:

x = 2

Question 3

Solve for

x

3^{2x} = 81

Solution:

Rewrite $81$ as $3^4$ : $3^{2x} = 3^4$ .
Equate the powers: $2x = 4$ .
Solve: $x = 2$ .

Answer:

x = 2

Question 4

Solve for

x

10^{x+2} = 1000

Solution:

Rewrite $1000$ as $10^3$ : $10^{x+2} = 10^3$ .
Equate the powers: $x + 2 = 3$ .
Solve: $x = 1$ .

Answer:

x = 1

Question 5

Solve for

x

7^{x-1} = 49

Solution:

Rewrite $49$ as $7^2$ : $7^{x-1} = 7^2$ .
Equate the powers: $x - 1 = 2$ .
Solve: $x = 3$ .

Answer:

x = 3

Question 6

Solve for

x

2^{x+3} = 2^5

Solution:

Equate the powers: $x + 3 = 5$ .
Solve: $x = 2$ .

Answer:

x = 2

Question 7

Solve for

x

9^x = 27^{x-1}

Solution:

Rewrite $9$ as $3^2$ and $27$ as $3^3$ : $(3^2)^x = (3^3)^{x-1}$ .
Simplify: $3^{2x} = 3^{3x - 3}$ .
Equate the powers: $2x = 3x - 3$ .
Solve: $x = 3$ .

Answer:

x = 3

Question 8

Solve for

x

4^{x+2} = 64

Solution:

Rewrite $64$ as $4^3$ : $4^{x+2} = 4^3$ .
Equate the powers: $x + 2 = 3$ .
Solve: $x = 1$ .

Answer:

x = 1

Question 9

Solve for

x

3^{x-1} = \frac{1}{27}

Solution:

Rewrite $\frac{1}{27}$ as $3^{-3}$ : $3^{x-1} = 3^{-3}$ .
Equate the powers: $x - 1 = -3$ .
Solve: $x = -2$ .

Answer:

x = -2

Question 10

Solve for

x

5^x \cdot 5^3 = 125

Solution:

Combine: $5^{x+3} = 125$ .
Rewrite $125$ as $5^3$ : $5^{x+3} = 5^3$ .
Equate the powers: $x + 3 = 3$ .
Solve: $x = 0$ .

Answer:

x = 0

Question 11

Solve for

x

2^{2x} = 32

Solution:

Rewrite $32$ as $2^5$ : $2^{2x} = 2^5$ .
Equate the powers: $2x = 5$ .
Solve: $x = \frac{5}{2}$ .

Answer:

x = \frac{5}{2}

Question 12

Solve for

x

6^{x-2} = 36

Solution:

Rewrite $36$ as $6^2$ : $6^{x-2} = 6^2$ .
Equate the powers: $x - 2 = 2$ .
Solve: $x = 4$ .

Answer:

x = 4

Question 13

Solve for

x

8^{x+1} = 64

Solution:

Rewrite $64$ as $8^2$ : $8^{x+1} = 8^2$ .
Equate the powers: $x + 1 = 2$ .
Solve: $x = 1$ .

Answer:

x = 1

Question 14

Solve for

x

2^{x+3} = 16 \cdot 2^x

Solution:

Rewrite $16$ as $2^4$ : $2^{x+3} = 2^4 \cdot 2^x$ .
Simplify: $2^{x+3} = 2^{x+4}$ .
Equate the powers: $x + 3 = x + 4$ .
No solution exists.

Answer: No solution.

Question 15

Solve for

x

9^x = \frac{1}{81}

Solution:

Rewrite $81$ as $9^2$ : $9^x = 9^{-2}$ .
Equate the powers: $x = -2$ .

Answer:

x = -2

Question 16

Solve for

x

(2^x)^2 = 64

Solution:

Simplify: $2^{2x} = 64$ .
Rewrite $64$ as $2^6$ : $2^{2x} = 2^6$ .
Equate the powers: $2x = 6$ .
Solve: $x = 3$ .

Answer:

x = 3

Question 17

Solve for

x

3^{2x} \cdot 3^4 = 27

Solution:

Rewrite $27$ as $3^3$ : $3^{2x+4} = 3^3$ .
Equate the powers: $2x + 4 = 3$ .
Solve: $2x = -1$ , $x = -\frac{1}{2}$ .

Answer:

x = -\frac{1}{2}

Question 18

Solve for

x

\frac{4^x}{2^x} = 8

Solution:

Rewrite $4^x$ as $(2^2)^x$ : $\frac{(2^2)^x}{2^x} = 8$ .
Simplify: $\frac{2^{2x}}{2^x} = 2^x$ .
Solve: $2^x = 8 = 2^3$ .
Equate: $x = 3$ .

Answer:

x = 3

Question 19

Solve for

x

5^x = 625

Solution:

Rewrite $625$ as $5^4$ : $5^x = 5^4$ .
Equate the powers: $x = 4$ .

Answer:

x = 4

Question 20

Solve for

x

2^{x-2} = 32

Solution:

Rewrite $32$ as $2^5$ : $2^{x-2} = 2^5$ .
Equate the powers: $x - 2 = 5$ .
Solve: $x = 7$ .

Answer:

x = 7

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Calculation problems on standard form

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Question 1

Write

450,000

in standard form.

Solution:

Express $450,000$ as $4.5 \times 10^5$ .
Explanation: Move the decimal point 5 places to the left.

Answer:

4.5 \times 10^5

Question 2

Convert

0.00032

to standard form.

Solution:

Rewrite $0.00032$ as $3.2 \times 10^{-4}$ .
Explanation: Move the decimal point 4 places to the right.

Answer:

3.2 \times 10^{-4}

Question 3

Simplify and write

(2 \times 10^3) \times (3 \times 10^2)

in standard form.

Solution:

Multiply: $2 \times 3 = 6$ and $10^3 \times 10^2 = 10^{3+2} = 10^5$ .
Final Answer: $6 \times 10^5$ .

Answer:

6 \times 10^5

Question 4

Express

\frac{6 \times 10^4}{2 \times 10^2}

in standard form.

Solution:

Divide: $6 \div 2 = 3$ and $10^4 \div 10^2 = 10^{4-2} = 10^2$ .
Final Answer: $3 \times 10^2$ .

Answer:

3 \times 10^2

Question 5

Write

7.2 \times 10^{-3}

as a decimal number.

Solution:

Move the decimal point 3 places to the left: $0.0072$ .

Answer:

0.0072

Question 6

Simplify:

(5 \times 10^6) + (3 \times 10^6)

Solution:

Combine: $5 + 3 = 8$ .
Result: $8 \times 10^6$ .

Answer:

8 \times 10^6

Question 7

Convert

0.00000045

to standard form.

Solution:

Rewrite $0.00000045$ as $4.5 \times 10^{-7}$ .
Explanation: Move the decimal 7 places to the right.

Answer:

4.5 \times 10^{-7}

Question 8

Write

9,500,000

in standard form.

Solution:

Rewrite $9,500,000$ as $9.5 \times 10^6$ .

Answer:

9.5 \times 10^6

Question 9

Simplify

(8 \times 10^{-5}) \times (4 \times 10^2)

and write in standard form.

Solution:

Multiply: $8 \times 4 = 32$ .
Combine powers: $10^{-5} \times 10^2 = 10^{-3}$ .
Adjust: $32 = 3.2 \times 10^1$ , so $(3.2 \times 10^1) \times 10^{-3} = 3.2 \times 10^{-2}$ .

Answer:

3.2 \times 10^{-2}

Question 10

Convert

1.23 \times 10^4

to a standard number.

Solution:

Move the decimal point 4 places to the right: $12300$ .

Answer:

12300

Question 11

Express

0.0000008

in standard form.

Solution:

Rewrite $0.0000008$ as $8 \times 10^{-7}$ .

Answer:

8 \times 10^{-7}

Question 12

Simplify:

\frac{4.5 \times 10^6}{1.5 \times 10^2}

Solution:

Divide: $4.5 \div 1.5 = 3$ .
Combine powers: $10^6 \div 10^2 = 10^{6-2} = 10^4$ .
Result: $3 \times 10^4$ .

Answer:

3 \times 10^4

Question 13

Convert

650,000

to standard form.

Solution:

Rewrite $650,000$ as $6.5 \times 10^5$ .

Answer:

6.5 \times 10^5

Question 14

Simplify

(2 \times 10^3) - (1.5 \times 10^3)

Solution:

Subtract coefficients: $2 - 1.5 = 0.5$ .
Result: $0.5 \times 10^3 = 5 \times 10^2$ .

Answer:

5 \times 10^2

Question 15

Write

0.00047

in standard form.

Solution:

Rewrite $0.00047$ as $4.7 \times 10^{-4}$ .

Answer:

4.7 \times 10^{-4}

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Calculation problems involving laws of logarithms

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Question 1.

Simplify

\log_a(mn)

using the laws of logarithms.

Solution:

Using the product rule: $\log_a(mn) = \log_a m + \log_a n$ .

Question 2.

Simplify

\log_a\left(\frac{m}{n}\right)

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Solution:

Using the quotient rule: $\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n$ .

Question 3.

Simplify

\log_a(m^n)

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Solution:

Using the power rule: $\log_a(m^n) = n \cdot \log_a m$ .

Question 4.

\log_2 8 = x

, find

x

Solution:

Rewrite in exponential form: $2^x = 8$ .
Solve $x = 3$ since $2^3 = 8$ .

Question 5.

Express

\log_a b

in terms of

\log_b a

Solution:

Using the change of base formula: $\log_a b = \frac{1}{\log_b a}$ .

Question 6.

\log_5 25 = x

, find

x

Solution:

Rewrite in exponential form: $5^x = 25$ .
Solve $x = 2$ since $5^2 = 25$ .

Question 7.

Solve

\log_3 27 = x

Solution:

Rewrite in exponential form: $3^x = 27$ .
Solve $x = 3$ since $3^3 = 27$ .

Question 8.

Simplify

2 \log_a m + \log_a n

Solution:

Using the power rule: $2 \log_a m = \log_a(m^2)$ .
Combine: $\log_a(m^2) + \log_a n = \log_a(m^2 \cdot n)$ .

Question 9.

Simplify

\log_a m - \log_a n + \log_a p

Solution:

Combine terms: $\log_a m - \log_a n = \log_a\left(\frac{m}{n}\right)$ .
Add $\log_a p$ : $\log_a\left(\frac{m}{n}\right) + \log_a p = \log_a\left(\frac{m \cdot p}{n}\right)$ .

Question 10.

Expand

\log_a\left(\frac{x^3y^2}{z}\right)

Solution:

Using the quotient rule: $\log_a\left(\frac{x^3y^2}{z}\right) = \log_a(x^3y^2) - \log_a z$ .
Using the product rule: $\log_a(x^3y^2) = \log_a(x^3) + \log_a(y^2)$ .
Using the power rule: $\log_a(x^3) = 3 \log_a x$ and $\log_a(y^2) = 2 \log_a y$ .
Combine: $3 \log_a x + 2 \log_a y - \log_a z$ .

Question 11.

Solve

\log_x 81 = 4

Solution:

Rewrite in exponential form: $x^4 = 81$ .
Solve $x = 3$ since $3^4 = 81$ .

Question 12.

Simplify

\log_a a^5

Solution:

Using the power rule: $\log_a a^5 = 5 \cdot \log_a a$ .
Since $\log_a a = 1$ , the result is $5$ .

Question 13.

Simplify

\log_2 32 - \log_2 8

Solution:

Using the quotient rule: $\log_2 32 - \log_2 8 = \log_2\left(\frac{32}{8}\right)$ .
Simplify $\frac{32}{8} = 4$ : $\log_2 4 = 2$ .

Question 14.

Solve

\log_10 x = 2

Solution:

Rewrite in exponential form: $10^2 = x$ .
Solve $x = 100$ .

Question 15.

Express

\log_2 5

in terms of

\log_{10} 2

and

\log_{10} 5

Solution:

Using the change of base formula: $\log_2 5 = \frac{\log_{10} 5}{\log_{10} 2}$ .

Question 16.

Solve

\log_3 x + \log_3(x - 2) = 1

Solution:

Combine logs: $\log_3[x(x - 2)] = 1$ .
Rewrite in exponential form: $3^1 = x(x - 2)$ .
Expand: $3 = x^2 - 2x$ .
Solve $x^2 - 2x - 3 = 0$ : $(x - 3)(x + 1) = 0$ .
Solutions: $x = 3$ (valid), $x = -1$ (invalid).

Question 17.

Simplify

3 \log_a x - \frac{1}{2} \log_a y

Solution:

Using the power rule: $3 \log_a x = \log_a(x^3)$ , $\frac{1}{2} \log_a y = \log_a(y^{1/2})$ .
Combine: $\log_a(x^3) - \log_a(y^{1/2})$ = $\log_a\left(\frac{x^3}{\sqrt{y}}\right)$ .

Question 18.

\log_a b = 2

and

\log_a c = 3

, find

\log_a(b^2c^3)

Solution:

Using the power rule: $\log_a(b^2c^3) = 2 \log_a b + 3 \log_a c$ .
Substitute: $2(2) + 3(3) = 4 + 9 = 13$ .

Question 19.

Solve

2 \log_2 x = 5

Solution:

Simplify: $\log_2 x = \frac{5}{2}$ .
Rewrite in exponential form: $x = 2^{5/2}$ = $\sqrt{32}$ = $4\sqrt{2}$ .

Question 20.

Simplify

\log_2 64 + \log_2 4 - \log_2 8

Solution:

Combine terms: $\log_2 64 + \log_2 4 = \log_2(64 \cdot 4) = \log_2 256$ .
Subtract $\log_2 8$ : $\log_2 256 - \log_2 8 = \log_2\left(\frac{256}{8}\right)$ .
Simplify $\frac{256}{8} = 32$ : $\log_2 32 = 5$ .
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Calculation problems involving change of bases in logarithm and application

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Question 1.

Rewrite

\log_5 20

using the change of base formula.

Solution:

Using the change of base formula: $\log_a b = \frac{\log_c b}{\log_c a}$ .
Rewrite: $\log_5 20 = \frac{\log_{10} 20}{\log_{10} 5}$ .

Question 2.

Simplify

\log_3 27

using the change of base formula

Solution:

Rewrite using the change of base formula: $\log_3 27 = \frac{\log_{10} 27}{\log_{10} 3}$ .
Simplify: $\frac{\log_{10}(3^3)}{\log_{10} 3} = \frac{3 \cdot \log_{10} 3}{\log_{10} 3} = 3$ .

Question 3.

\log_2 10 \approx 3.32

, approximate

\log_4 10

Solution:

Use the formula $\log_a b = \frac{\log_c b}{\log_c a}$ .
Rewrite: $\log_4 10 = \frac{\log_2 10}{\log_2 4}$ .
Substitute values: $\log_2 4 = 2$ , so $\log_4 10 = \frac{3.32}{2} \approx 1.66$ .

Questjion 4.

Simplify

\log_7 343

using the change of base formula.

Solution:

Rewrite: $\log_7 343 = \frac{\log_{10} 343}{\log_{10} 7}$ .
Simplify: $\log_{10}(7^3) = \frac{3 \cdot \log_{10} 7}{\log_{10} 7} = 3$ .

Question 5.

Find

\log_3 5

using

\log_{10} 3 = 0.477

and

\log_{10} 5 = 0.699

Solution:

Use the formula $\log_a b = \frac{\log_c b}{\log_c a}$ .
Rewrite: $\log_3 5 = \frac{\log_{10} 5}{\log_{10} 3}$ .
Substitute values: $\log_3 5 = \frac{0.699}{0.477} \approx 1.466$ .

Question 6.

Simplify

\log_6 36

using the change of base formula.

Solution:

Rewrite: $\log_6 36 = \frac{\log_{10} 36}{\log_{10} 6}$ .
Simplify: $\frac{\log_{10}(6^2)}{\log_{10} 6} = \frac{2 \cdot \log_{10} 6}{\log_{10} 6} = 2$ .

Question 7.

\log_{10} 2 = 0.301

, find

\log_5 2

Solution:

Use $\log_a b = \frac{\log_c b}{\log_c a}$ .
Rewrite: $\log_5 2 = \frac{\log_{10} 2}{\log_{10} 5}$ .
Substitute: $log_10 5 = 0.699 $, so$ \log_5 2 = \frac0.3010.699 \approx 0.431 $.

Question 8.

Find

\log_8 64

using the change of base formula.

Solution:

Rewrite: $\log_8 64 = \frac{\log_{10} 64}{\log_{10} 8}$ .
Simplify: $\log_{10}(8^2) = \frac{2 \cdot \log_{10} 8}{\log_{10} 8} = 2$

Question 9.

Simplify

\log_5 25

using change of base.

Solution:

Rewrite: $\log_5 25 = \frac{\log_{10} 25}{\log_{10} 5}$ .
Simplify: $\frac{\log_{10}(5^2)}{\log_{10} 5} = \frac{2 \cdot \log_{10} 5}{\log_{10} 5} = 2$ .

Question 10.

Use the change of base formula to rewrite $ \log_4 32 .

Solution:

Rewrite: $\log_4 32 = \frac{\log_{10} 32}{\log_{10} 4}$ .

Question 11.

Solve

\log_3 9

.**

Solution:

Rewrite: $\log_3 9 = \frac{\log_{10} 9}{\log_{10} 3}$ .
Simplify: $\frac{\log_{10}(3^2)}{\log_{10} 3} = \frac{2 \cdot \log_{10} 3}{\log_{10} 3} = 2$ .

Question 12.

Evaluate

\log_2 32

using the change of base formula.

Solution:

Rewrite: $\log_2 32 = \frac{\log_{10} 32}{\log_{10} 2}$ .
Simplify: $32 = 2^5$ , so $\frac{5 \cdot \log_{10} 2}{\log_{10} 2} = 5$ .

Question 13.

Approximate

\log_3 81

using

\log_{10} 3 = 0.477

and

\log_{10} 81 = 1.908

Solution:

Rewrite: $\log_3 81 = \frac{\log_{10} 81}{\log_{10} 3}$ .
Substitute values: $\frac{1.908}{0.477} = 4$ .

Question 14.

\log_{10} 2 = 0.301

, find

\log_4 2

Solution:

Rewrite: $\log_4 2 = \frac{\log_{10} 2}{\log_{10} 4}$ .
Substitute: $\log_{10} 4 = 2 \cdot \log_{10} 2 = 0.602$ .
Solve: $\frac{0.301}{0.602} = 0.5$ .

Question 15.

Simplify

\log_7 49

Solution:

Rewrite: $\log_7 49 = \frac{\log_{10} 49}{\log_{10} 7}$ .
Simplify: $\log_{10}(7^2) = \frac{2 \cdot \log_{10} 7}{\log_{10} 7} = 2$ .

Question 16.

Rewrite

\log_8 16

using the change of base formula.

Solution:

Rewrite: $\log_8 16 = \frac{\log_{10} 16}{\log_{10} 8}$ .

Question 17.

Approximate

\log_3 10

using

\log_{10} 3 = 0.477

and

\log_{10} 10 = 1

Solution:

Rewrite: $\log_3 10 = \frac{\log_{10} 10}{\log_{10} 3}$ .
Substitute: $\frac{1}{0.477} \approx 2.096$ .

Question 18.

Solve

\log_4 64

Solution:

Rewrite: $\log_4 64 = \frac{\log_{10} 64}{\log_{10} 4}$ .
Simplify: $64 = 4^3$ , so $\frac{3 \cdot \log_{10} 4}{\log_{10} 4} = 3$ .

Question 19.

Find

\log_6 216

Solution:

Rewrite: $\log_6 216 = \frac{\log_{10} 216}{\log_{10} 6}$ .
Simplify: $216 = 6^3$ , so $\log_6 216 = 3$ .

Question 20.

Simplify

\log_5 125

Solution:

Rewrite: $\log_5 125 = \frac{\log_{10} 125}{\log_{10} 5}$ .
Simplify: $\frac{\log_{10}(5^3)}{\log_{10} 5} = \frac{3 \cdot \log_{10} 5}{\log_{10} 5} = 3$ .
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Calculation problems involving simplification and rationalization of surds

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Question 1.

Simplify

\sqrt{50}

Solution:

Factorize $50 = 25 \cdot 2$ .
$\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2}$ .
$\sqrt{50} = 5\sqrt{2}$ .

Question 2.

Simplify

\sqrt{72}

Solution:

Factorize $72 = 36 \cdot 2$ .
$\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2}$ .
$\sqrt{72} = 6\sqrt{2}$ .

Question 3.

Rationalize

\frac{1}{\sqrt{3}}

Solution:

Multiply numerator and denominator by $\sqrt{3}$ : $\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$ .
Result: $\frac{\sqrt{3}}{3}$ .

Question 4.

Simplify

\sqrt{32} + \sqrt{50}

Solution:

$\sqrt{32} = 4\sqrt{2}$ , $\sqrt{50} = 5\sqrt{2}$ .
Combine: $4\sqrt{2} + 5\sqrt{2} = 9\sqrt{2}$ .

Question 5.

Simplify

\sqrt{12} - \sqrt{27}

Solution:

$\sqrt{12} = 2\sqrt{3}$ , $\sqrt{27} = 3\sqrt{3}$ .
Combine: $2\sqrt{3} - 3\sqrt{3} = -\sqrt{3}$ .

Question 6.

Rationalize

\frac{2}{\sqrt{5}}

Solution:

Multiply numerator and denominator by $\sqrt{5}$ : $\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$ .
Result: $\frac{2\sqrt{5}}{5}$ .

Question 7.

Rationalize

\frac{1}{\sqrt{7} + 2}

Solution:

Multiply numerator and denominator by the conjugate: $(\sqrt{7} - 2)$ .
Numerator: $1 \cdot (\sqrt{7} - 2) = \sqrt{7} - 2$ .
Denominator: $(\sqrt{7} + 2)(\sqrt{7} - 2) = 7 - 4 = 3$ .
Result: $\frac{\sqrt{7} - 2}{3}$ .

Question 8.

Simplify

\sqrt{18} \cdot \sqrt{8}

Solution:

Combine: $\sqrt{18} \cdot \sqrt{8} = \sqrt{18 \cdot 8} = \sqrt{144}$ .
Result: $\sqrt{144} = 12$ .

Question 9.

Rationalize

\frac{\sqrt{2}}{\sqrt{3}}

.**

Solution:

Multiply numerator and denominator by $\sqrt{3}$ : $\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$ .
Result: $\frac{\sqrt{6}}{3}$ .

Question 10.

Simplify

\sqrt{75} + \sqrt{27}

Solution:

$\sqrt{75} = 5\sqrt{3}$ , $\sqrt{27} = 3\sqrt{3}$ .
Combine: $5\sqrt{3} + 3\sqrt{3} = 8\sqrt{3}$ .

Question 11.

Rationalize

\frac{1}{\sqrt{2} - 1}

Solution:

Multiply numerator and denominator by the conjugate: $(\sqrt{2} + 1)$ .
Numerator: $1 \cdot (\sqrt{2} + 1) = \sqrt{2} + 1$ .
Denominator: $(\sqrt{2} - 1)(\sqrt{2} + 1) = 2 - 1 = 1$ .
Result: $\sqrt{2} + 1$ .

Question 12.

Simplify

\sqrt{20} \cdot \sqrt{45}

Solution:

Combine: $\sqrt{20} \cdot \sqrt{45} = \sqrt{20 \cdot 45} = \sqrt{900}$ .
Result: $\sqrt{900} = 30$ .

Question 13.

Simplify

\sqrt{8} + \sqrt{32}

Solution:

$\sqrt{8} = 2\sqrt{2}$ , $\sqrt{32} = 4\sqrt{2}$ .
Combine: $2\sqrt{2} + 4\sqrt{2} = 6\sqrt{2}$ .

Question 14.

Rationalize

\frac{\sqrt{3}}{\sqrt{5} - \sqrt{2}}

Solution:

Multiply numerator and denominator by the conjugate: $(\sqrt{5} + \sqrt{2})$ .
Numerator: $\sqrt{3}(\sqrt{5} + \sqrt{2}) = \sqrt{15} + \sqrt{6}$ .
Denominator: $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = 5 - 2 = 3$ .
Result: $\frac{\sqrt{15} + \sqrt{6}}{3}$ .

Question 15.

Simplify

\frac{\sqrt{50}}{\sqrt{2}}

Solution:

Combine: $\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25}$ .
Result: $\sqrt{25} = 5$ .

Question 16.

Rationalize

\frac{2}{\sqrt{6}}

Solution:

Multiply numerator and denominator by $\sqrt{6}$ : $\frac{2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$ .
Result: $\frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3}$ .

Question 17.

Simplify

\sqrt{27} \cdot \sqrt{12}

Solution:

Combine: $\sqrt{27} \cdot \sqrt{12} = \sqrt{27 \cdot 12} = \sqrt{324}$ .
Result: $\sqrt{324} = 18$ .

Question 18.

Rationalize

\frac{3}{\sqrt{7} + 2}

Solution:

Multiply numerator and denominator by the conjugate: $(\sqrt{7} - 2)$ .
Numerator: $3(\sqrt{7} - 2) = 3\sqrt{7} - 6$ .
Denominator: $(\sqrt{7} + 2)(\sqrt{7} - 2) = 7 - 4 = 3$ .
Result: $\sqrt{7} - 2$ .

Question 19.

Simplify

\frac{\sqrt{45}}{\sqrt{15}}

.**

Solution:

Combine: $\frac{\sqrt{45}}{\sqrt{15}} = \sqrt{\frac{45}{15}} = \sqrt{3}$ .
Result: $\sqrt{3}$ .

Question 20.

Rationalize

\frac{\sqrt{5} - 1}{\sqrt{5} + 2}

.**

Solution:

Multiply numerator and denominator by the conjugate: $(\sqrt{5} - 2)$ .
Numerator: $(\sqrt{5} - 1)(\sqrt{5} - 2) = 5 - 2\sqrt{5} - \sqrt{5} + 2 = 7 - 3\sqrt{5}$ .
Denominator: $(\sqrt{5} + 2)(\sqrt{5} - 2) = 5 - 4 = 1$ .
Result: $7 - 3\sqrt{5}$ .

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- Jamb Mathematics- Lesson notes on Sets for utme Success

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This is all we can take on "Jamb Mathematics - Lesson Notes on Indices, Logarithms and surds for UTME Candidate"

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