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

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

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

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

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

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

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

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

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

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Question 2: Find the value of \displaystyle  x :

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

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

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

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

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

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

(vii) \displaystyle  \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 \displaystyle  x :

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

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

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

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

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

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

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

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

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Question 5: Find the value:

(i) \displaystyle  \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 \displaystyle  \log_2 y = x , find the value of \displaystyle  8^x in terms of \displaystyle  y .

Answer:

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

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

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

Answer:

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

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

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

Answer:

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

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

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

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

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

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

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

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

\displaystyle  = \log_{10} \frac{x^2 \sqrt{y}}{z^3}  

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

Answer:

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

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

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

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

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

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

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

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

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

Answer:

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

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

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

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

Answer:

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

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

\displaystyle \text{Therefore  }   xy = 10^a. 10^b = 10^{a+b}

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

Answer:

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

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

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

Answer:

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

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

\displaystyle \text{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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