1. Questions, Correct Options, and Reasons
(i) The standard form of \(5.2 \times 10^6\) is:
(a) 52,000
(b) 520,000
(c) 5,200,000
(d) 52,000,000
Correct Answer: (c) 5,200,000
Reason: Multiplying \(5.2\) by \(10^6\) gives \(5,200,000\).
(ii) Scientific notation of \(0.00034\) is:
(a) \(3.4 \times 10^3\)
(b) \(3.4 \times 10^{-4}\)
(c) \(3.4 \times 10^4\)
(d) \(3.4 \times 10^{-3}\)
Correct Answer: (b) \(3.4 \times 10^{-4}\)
Reason: Moving the decimal 4 places to the right gives \(3.4 \times 10^{-4}\).
(iii) The base of common logarithm is:
(a) 2
(b) 10
(c) 5
(d) \(e\)
Correct Answer: (b) 10
Reason: The common logarithm (log) is based on \(10\).
(iv) \(\log_2 2^3 = \_\_\_\_\_\_\_\_\_\_\_\_\).
(a) 1
(b) 2
(c) 5
(d) 3
Correct Answer: (d) 3
Reason: \(\log_2 2^3 = 3\), as \(2^3 = 8\).
(v) \(\log 100 = \_\_\_\_\_\_\_\_\_\_\_\_\).
(a) 2
(b) 3
(c) 10
(d) 1
Correct Answer: (a) 2
Reason: \(\log 100 = \log 10^2 = 2\).
(vi) If \(\log 2 = 0.3010\), then \(\log 200\) is:
(a) 1.3010
(b) 0.6010
(c) 2.3010
(d) 2.6010
Correct Answer: (c) 2.3010
Reason: \(\log 200 = \log (2 \times 10^2) = \log 2 + \log 10^2 = 0.3010 + 2 = 2.3010\).
(vii) \(\log(0) = \_\_\_\_\_\_\_\_\_\_\_\_\).
(a) Positive
(b) Negative
(c) Zero
(d) Undefined
Correct Answer: (d) Undefined
Reason: Logarithm of \(0\) is undefined because no power of a base can equal \(0\).
(viii) \(\log 10,000 = \_\_\_\_\_\_\_\_\_\_\_\_\).
(a) 2
(b) 3
(c) 4
(d) 5
Correct Answer: (c) 4
Reason: \(\log 10,000 = \log(10^4) = 4\).
(ix) \(\log 5 + \log 3 = \_\_\_\_\_\_\_\_\_\_\_\_\).
(a) \(\log 0\)
(b) \(\log 2\)
(c) \(\log \left(\frac{5}{3}\right)\)
(d) \(\log 15\)
Correct Answer: (d) \(\log 15\)
Reason: \(\log 5 + \log 3 = \log(5 \times 3) = \log 15\).
(x) \(3^4 = 81\) in logarithmic form is:
(a) \(\log_3 4 = 81\)
(b) \(\log_4 3 = 81\)
(c) \(\log_3 81 = 4\)
(d) \(\log_4 81 = 3\)
Correct Answer: (c) \(\log_3 81 = 4\)
Reason: The logarithmic form of \(3^4 = 81\) is \(\log_3 81 = 4\).
2. Express the following numbers in scientific notation:
(i) \(0.000567\)
\[0.000567 = 5.67 \times 10^{-4}\]
(ii) \(734\)
\[734 = 7.34 \times 10^2\]
(iii) \(0.33 \times 10^3\)
\[0.33 \times 10^3 = 3.3 \times 10^2\]
3. Express the following numbers in ordinary notation:
(i) \(2.6 \times 10^3\)
\[2.6 \times 10^3 = 2600\]
(ii) \(8.794 \times 10^{-4}\)
\[8.794 \times 10^{-4} = 0.0008794\]
(iii) \(6 \times 10^{-6}\)
\[6 \times 10^{-6} = 0.000006\]
4. Express each of the following in logarithmic form:
(i) \(3^7 = 2187\)
\[\log_3 2187 = 7\]
(ii) \(a^b = c\)
\[\log_a c = b\]
(iii) \((12)^2 = 144\)
\[\log_{12} 144 = 2\]
5. Express each of the following in exponential form:
(i) \(\log_4 8 = x\)
\[4^x = 8\]
(ii) \(\log_9 729 = 3\)
\[9^3 = 729\]
(iii) \(\log_4 1024 = 5\)
\[4^5 = 1024\]
6. Find value of \(x\) in the following:
(i) \(\log_9 x = 0.5\)
\[x = 9^{0.5} = 3\]
(ii) \(\left(\frac{1}{9}\right)^{3x} = 27\)
\[\frac{1}{9} = 3^{-2}, \, 27 = 3^3\]
\[(3^{-2})^{3x} = 3^3\]
\[3^{-6x} = 3^3\]
\[-6x = 3 \]
\[ x = -\frac{1}{2}\]
(iii) \(\left(\frac{1}{32}\right)^{2x} = 64\)
\[\frac{1}{32} = 2^{-5}, \, 64 = 2^6\]
\[(2^{-5})^{2x} = 2^6\]
\[2^{-10x} = 2^6\]
\[-10x = 6 \]
\[ x = -\frac{3}{5}\]
7. Write the following as a single logarithm:
(i) \(7 \log x – 3 \log y^2\)
\[7 \log x – 3 \log y^2 = \log(x^7) – \log(y^6)\]
\[= \log\left(\frac{x^7}{y^6}\right)\]
(ii) \(3 \log 4 – \log 32\)
\[3 \log 4 – \log 32 = \log(4^3) – \log(32)\]
\[= \log\left(\frac{4^3}{32}\right) = \log\left(\frac{64}{32}\right) = \log(2)\]
(iii) \(\frac{1}{3}(\log_5 8 + \log_5 27) – \log_5 3\)
\[\frac{1}{3}(\log_5 8 + \log_5 27) – \log_5 3 = \frac{1}{3} \log_5(8 \cdot 27) – \log_5 3\]
\[= \frac{1}{3} \log_5(216) – \log_5 3 = \log_5(216^{1/3}) – \log_5 3\]
\[= \log_5(6) – \log_5(3) = \log_5\left(\frac{6}{3}\right) = \log_5(2)\]
8. Expand the following using laws of logarithms:
(i) \(\log(x y z^6)\)
\[\log(x y z^6) = \log(x) + \log(y) + 6\log(z)\]
(ii) \(\log_3 \sqrt[6]{m^5 n^3}\)
\[\log_3 \sqrt[6]{m^5 n^3} = \frac{1}{6} \log_3(m^5 n^3)\]
\[= \frac{1}{6} (\log_3(m^5) + \log_3(n^3))\]
\[= \frac{1}{6} (5 \log_3 m + 3 \log_3 n)\]
\[= \frac{5}{6} \log_3 m + \frac{3}{6} \log_3 n\]
\[= \frac{5}{6} \log_3 m + \frac{1}{2} \log_3 n\]
(iii) \(\log\sqrt{8x^3}\)
\[\log\sqrt{8x^3} = \frac{1}{2} \log(8x^3)\]
\[= \frac{1}{2} (\log(8) + \log(x^3))\]
\[= \frac{1}{2} (\log(8) + 3 \log(x))\]
\[= \frac{\log(8)}{2} + \frac{3}{2} \log(x)\]
9. Find the values of the following with the help of logarithm table:
(i) \(\sqrt[3]{68.24}\)
\[\sqrt[3]{68.24} = (68.24)^{1/3}\]
\[\log(68.24) = 1.8330, \, \frac{1.8330}{3} = 0.6110\]
\[10^{0.6110} = 4.1\]
(ii) \(319.8 \times 3.543\)
\[\log(319.8) = 2.5049, \, \log(3.543) = 0.5497\]
\[\log(\text{Result}) = 2.5049 + 0.5497 = 3.0546\]
\[10^{3.0546} = 1133\]
(iii) \(\frac{36.12 \times 750.9}{113.2 \times 9.98}\)
\[\log(36.12) = 1.5586, \, \log(750.9) = 2.8769\]
\[\log(113.2) = 2.0538, \, \log(9.98) = 0.9996\]
\[\log(\text{Result}) = (1.5586 + 2.8769) – (2.0538 + 0.9996)\]
\[= 4.4355 – 3.0534 = 1.3821\]
\[10^{1.3821} = 24.1\]
10. Population Growth
Given:
\[p(t) = 22(1.025)^t, \, p(t) = 35\]
\[35 = 22(1.025)^t\]
\[\frac{35}{22} = (1.025)^t\]
\[\log(1.5909) = t \cdot \log(1.025)\]
\[\log(1.5909) = 0.2014, \, \log(1.025) = 0.0107\]
\[t = \frac{0.2014}{0.0107} = 18.82\]
\[\text{Year} = 2016 + 19 = 2035\]
