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 $y$.

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

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 $x$.

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

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 $z$.

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

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

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

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 $y$.

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

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

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

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

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

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

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

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 $y$.

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

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

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 $xy$.

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

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

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 $x$.

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

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 $y$.

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

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

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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