Question 1: Write the following in the from of logarithms:

(i) 2^6 = 64 \Longleftrightarrow \log_2 64 = 6

(ii) 10^4 = 10000 \Longleftrightarrow \log_{10} 10000 = 4

(iii) 3^5 = 243 \Longleftrightarrow \log_3 243 = 5

(iv) 3^{-3} = \frac{1}{27} \Longleftrightarrow \log_3 {\frac{1}{27}} = -3

(v) 10^{-3} = 0.001 \Longleftrightarrow \log_{10} 0.001 = -3

(vi) 7^2 = 49 \Longleftrightarrow \log_7 49 = 2

(vii) 2^{-6} = \frac{1}{64} \Longleftrightarrow \log_2 {\frac{1}{64}} = -6

(viii) 2^{\frac{3}{2}} = 8 \Longleftrightarrow \log_4 8 = \frac{3}{2}

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Question 2: Find the value of x :

(i) \log_3 x = 4 \hspace{0.5cm} \Rightarrow x = 3^4  \hspace{0.5cm}  \Rightarrow x = 81

(ii) \log_4 x = 3 \hspace{0.5cm} \Rightarrow x = 4^3  \hspace{0.5cm}  \Rightarrow x = 64

(iii) \log_{\sqrt{3}} x = 4 \hspace{0.5cm} \Rightarrow x = (\sqrt{3})^4  \hspace{0.5cm}  \Rightarrow x = 9

(iv) \log_{10} x = -3 \hspace{0.5cm} \Rightarrow x = 10^{-3}  \hspace{0.5cm}  \Rightarrow x = 0.001

(v) \log_4 x = 1.5 \hspace{0.5cm} \Rightarrow x = 4^{1.5}  \hspace{0.5cm}  \Rightarrow x = 8

(vi) \log_8 x = \frac{2}{3} \hspace{0.5cm} \Rightarrow x = 8^{\frac{2}{3}} \hspace{0.5cm}  \Rightarrow x = 4

(vii)  \log_{125} x = \frac{1}{6} \hspace{0.5cm} \Rightarrow x = 125^{\frac{1}{6}}  \hspace{0.5cm}  \Rightarrow x = \sqrt{5}

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Question 3: Solve for x :

(i) \log_{\sqrt{5}} x = 4 \hspace{0.5cm} \Rightarrow (\sqrt{5})^4 = x \hspace{0.5cm} \Rightarrow x = 25

(ii) \log_{x} 0.0001 = -4 \hspace{0.5cm} \Rightarrow x^{-4} = 0.0001 \hspace{0.5cm} \Rightarrow x = 10

(iii) \log_{\sqrt{3}} (x-1) = 2 \hspace{0.5cm} \Rightarrow (\sqrt{3})^2 = (x-1) \hspace{0.5cm} \Rightarrow x = 4

(iv) \log_{3} (x^2+5) = 2 \hspace{0.5cm} \Rightarrow 3^{2} = x^2+5 \hspace{0.5cm} \Rightarrow x = \pm 2

(v) \log_{10} (3x-2) = 1 \hspace{0.5cm} \Rightarrow 10^{1} = 3x-2 \hspace{0.5cm} \Rightarrow x = 4

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Question 4: Express the following in exponential form:

(i) \log_5 25 = 2 \hspace{0.5cm} \Rightarrow 5^2 = 25

(ii) \log_4 64 = 3 \hspace{0.5cm} \Rightarrow 4^3 = 64

(iii) \log_{10} 0.001 = -3 \hspace{0.5cm} \Rightarrow 10^{-3} = 0.001

(iv) \log_{10} 1000 = 3 \hspace{0.5cm} \Rightarrow 10^3 = 1000

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Question 5: Find the value:

(i) \log_{\sqrt{3}} 3\sqrt{3} - \log_5 (0.25) = \log_{\sqrt{3}} 3\sqrt{3} - \log_5 (0.5)^2 = 3 + 2 \log_5 5 = 5

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Question 6: If \log_2 y = x , find the value of 8^x in terms of y .

Answer:

\log_2 y = x \hspace{0.5cm} \Rightarrow 2^x = y 

Therefore 2^{3x} = y^3 \hspace{0.5cm} \Rightarrow 8^x = y^3

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Question 7: If \log_{10} x = a , find the value of 10^{2a-1} in terms of x .

Answer:

\log_{10} x = a \hspace{0.5cm} \Rightarrow 10^a = x

Therefore 10^{2a-1} = \frac{10^{2a}}{10} = \frac{x^2}{10} = 0.1x^2

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Question 8: Given \log_{10} x = a, \ \log_{10} y = b and \log_{10} z = c , write down 10^{2a-1} in terms of x and 10^{3b-1} in terms of y . If \log_{10} u = 2a + \frac{b}{2} -3c , express u in terms of x, \ y and z .

Answer:

\log_{10} x = a \hspace{0.5cm} \Rightarrow 10^a = x

\log_{10} y = b  \hspace{0.5cm} \Rightarrow 10^b = y

\log_{10} z = c \hspace{0.5cm} \Rightarrow 10^c = z

Therefore 10^{2a-3} = \frac{1}{1000} . 10^{2a} = \frac{x^2}{1000} 

10^{3b-1} = \frac{1}{10} . 10^{3b} = \frac{y^3}{10} 

\log_{10} u = 2a + \frac{b}{2} -3c

= 2 \log_{10} x + \frac{1}{2} \log_{10} y - 3 \log_{10} z

=\log_{10} x^2 + \log_{10}y^{\frac{1}{2}} - \log_{10} z^3

= \log_{10} \frac{x^2 \sqrt{y}}{z^3} 

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Question 9: If \log_{10} x = 2a, \ 2\log_{10} y = b , then write 10^a in terms of x, 10^{2b-1} in terms of y . Also, if \log_{10} z = 3a - 2b , express z in terms of x and y .

Answer:

\log_{10} x = 2a \hspace{0.5cm} \Rightarrow 10^{2a} = x

2 \log_{10} y = b \hspace{0.5cm} \Rightarrow \log_{10} y^2 = b \hspace{0.5cm} \Rightarrow 10^b = y^2

10^a = (10^{2a})^{\frac{1}{2}} = x^{\frac{1}{2}}

10^{2b+1} = 10. 10^{2b} = 10. y^4

\log_{10} z = 3a - 2b = \frac{3}{2} \log_{10} x - 4 \log_{10} y

= \log_{10} x^{\frac{3}{2}} - \log_{10} y^4

= \log_{10} \frac{x^{\frac{3}{2}}}{y^4} 

Therefore z = \frac{x^{\frac{3}{4}}}{y^4} 

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Question 10: If x = \log_{10} a and y = \log_{10} b , express \frac{a^3}{b^2} in terms of x and y .

Answer:

\log_{10} a = x \hspace{0.5cm} \Rightarrow 10^x = a

\log_{10} b = y \hspace{0.5cm} \Rightarrow 10^y = b

Therefore \frac{a^3}{b^2} = \frac{10^{3x}}{10^{2y}} = 10^{3x-2y}

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Question 11: If \log_{10} x = a and \log_{10} y = b , find the value of xy .

Answer:

\log_{10} x = a \hspace{0.5cm} \Rightarrow 10^a = x

\log_{10} y = b \hspace{0.5cm} \Rightarrow 10^b = y

Therefore xy = 10^a. 10^b = 10^{a+b}

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Question 12: If \log_{10} x = y , express 10^{2y - 3} in terms of x .

Answer:

\log_{10} x = y \hspace{0.5cm} \Rightarrow 10^y = x

Therefore 10^{2y-3} = \frac{10^{2y}}{1000} = \frac{x^2}{1000}

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Question 13: If \log_2 x = a and \log_3 y = a , find 12^{2a-1} in terms of x and y .

Answer:

\log_2 x = a \hspace{0.5cm} \Rightarrow 2^a = x

\log_3 y = a \hspace{0.5cm} \Rightarrow 3^a = y

Therefore 12^{2a-1} = \frac{1}{12} (12^{2a}) = \frac{1}{12} (3 \times 4)^{2a} = \frac{1}{12} 3^{2a}. 4^{2a} = \frac{1}{12} y^2x^4

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