1. Without using a calculator, evaluate the following:
(i) \(\log_2 18 – \log_2 9\)
Using the logarithmic property: \(\log_b x – \log_b y = \log_b \frac{x}{y}\):
\[\log_2 18 – \log_2 9 = \log_2 \frac{18}{9} = \log_2 2 = 1\]
Answer: \(1\)
(ii) \(\log_2 64 + \log_2 2\)
Using the logarithmic property: \(\log_b x + \log_b y = \log_b (x \cdot y)\):
\[\log_2 64 + \log_2 2 = \log_2 (64 \cdot 2) = \log_2 128 = 7\]
Answer: \(7\)
(iii) \(\frac{1}{3} \log_3 8 – \log_3 18\)
Using the logarithmic property: \(k \cdot \log_b x = \log_b x^k\):
\[\frac{1}{3} \log_3 8 – \log_3 18 = \log_3 8^{1/3} – \log_3 18 = \log_3 2 – \log_3 18\]
\[= \log_3 \frac{2}{18} = \log_3 \frac{1}{9} = -2\]
Answer: \(-2\)
(iv) \(2 \log 2 + \log 25\)
Using the logarithmic property: \(k \cdot \log_b x = \log_b x^k\) and \(\log_b x + \log_b y = \log_b (x \cdot y)\):
\[2 \log 2 + \log 25 = \log 2^2 + \log 25 = \log 4 + \log 25 = \log (4 \cdot 25) = \log 100\]
\[\log 100 = 2\]
Answer: \(2\)
(v) \(\frac{1}{3} \log_4 64 + 2 \log_5 25\)
Using the logarithmic properties:
\[\frac{1}{3} \log_4 64 = \log_4 64^{1/3} = \log_4 4 = 1\]
\[2 \log_5 25 = \log_5 25^2 = \log_5 625 = 4\]
\[\text{Result: } 1 + 4 = 5\]
Answer: \(5\)
(vi) \(\log_{3} 12 + \log_{3} 0.25\)
Using the logarithmic property: \(\log_b x + \log_b y = \log_b (x \cdot y)\):
\[\log_{3} 12 + \log_{3} 0.25 = \log_{3} (12 \cdot 0.25) = \log_{3} 3\]
Answer: \(\log_{3} 3\)
2. Write the following as a single logarithm:
(i) \(\frac{1}{2} \log 25 + 2 \log 3\)
Using the logarithmic property: \(k \cdot \log_b x = \log_b x^k\):
\[\frac{1}{2} \log 25 + 2 \log 3 = \log 25^{1/2} + \log 3^2 = \log 5 + \log 9\]
\[\log 5 + \log 9 = \log (5 \cdot 9) = \log 45\]
Answer: \(\log 45\)
(ii) \(\log 9 – \log \frac{1}{3}\)
Using the logarithmic property: \(\log_b x – \log_b y = \log_b \frac{x}{y}\):
\[\log 9 – \log \frac{1}{3} = \log \frac{9}{\frac{1}{3}} = \log (9 \cdot 3) = \log 27\]
Answer: \(\log 27\)
(iii) \(\log_5 b^2 \cdot \log_a 5^3\)
Using the logarithmic property: \(k \cdot \log_b x = \log_b x^k\):
\[\log_5 b^2 \cdot \log_a 5^3 = 2 \log_5 b + 3 \log_a 5\]
Answer: \(2 \log_5 b + 3 \log_a 5\)
(iv) \(2 \log_3 x + \log_3 y\)
Using the logarithmic property: \(k \cdot \log_b x = \log_b x^k\):
\[2 \log_3 x + \log_3 y = \log_3 x^2 + \log_3 y = \log_3 (x^2 \cdot y)\]
Answer: \(\log_3 (x^2 \cdot y)\)
(v) \(4 \log_5 x – \log_5 y + \log_5 z\)
Using the logarithmic property:
\[4 \log_5 x – \log_5 y + \log_5 z = \log_5 x^4 – \log_5 y + \log_5 z = \log_5 \frac{x^4 \cdot z}{y}\]
Answer: \(\log_5 \frac{x^4 \cdot z}{y}\)
(vi) \(2 \ln a + 3 \ln b – 4 \ln c\)
Using the logarithmic property: \(k \cdot \ln x = \ln x^k\):
\[2 \ln a + 3 \ln b – 4 \ln c = \ln a^2 + \ln b^3 – \ln c^4 = \ln \frac{a^2 \cdot b^3}{c^4}\]
Answer: \(\ln \frac{a^2 \cdot b^3}{c^4}\)
3. Expand the following using the laws of logarithms:
(i) \(\log \left(\frac{11}{5}\right)\)
Using the logarithmic property: \(\log \frac{x}{y} = \log x – \log y\):
\[\log \left(\frac{11}{5}\right) = \log 11 – \log 5\]
Answer: \(\log 11 – \log 5\)
(ii) \(\log_5 \sqrt{8a^6}\)
\[\log_5 \sqrt{8a^6} = \log_5 (8a^6)^{1/2}\]
\[= \frac{1}{2} \log_5 (8a^6)\]
\[= \frac{1}{2} (\log_5 8 + \log_5 a^6)\]
\[= \frac{1}{2} (\log_5 8 + 6 \log_5 a)\]
\[= \frac{1}{2} \log_5 8 + 3 \log_5 a\]
(iii) \(\ln \left(\frac{a^2 b}{c}\right)\)
Using the logarithmic property:
\[\ln \left(\frac{a^2 b}{c}\right) = \ln a^2 + \ln b – \ln c = 2 \ln a + \ln b – \ln c\]
Answer: \(2 \ln a + \ln b – \ln c\)
(iv) \(\log \left(\frac{xy}{z}\right)^{\frac{1}{9}}\)
Using the logarithmic property:
\[\log \left(\frac{xy}{z}\right)^{\frac{1}{9}} = \frac{1}{9} \log \frac{xy}{z} = \frac{1}{9} [\log x + \log y – \log z]\]
Answer: \(\frac{1}{9} (\log x + \log y – \log z)\)
(v) \(\ln \sqrt[3]{16x^3}\)
Using the logarithmic property:
\[\ln \sqrt[3]{16x^3} = \frac{1}{3} \ln (16x^3) = \frac{1}{3} [\ln 16 + \ln x^3]\]
\[= \frac{1}{3} (\ln 16 + 3 \ln x)\]
Answer: \(\frac{1}{3} \ln 16 + \ln x\)
(vi) \(\log_2 \left(\frac{1 – a}{b}\right)^5\)
Using the logarithmic property:
\[\log_2 \left(\frac{1 – a}{b}\right)^5 = 5 \log_2 \frac{1 – a}{b} = 5 (\log_2 (1 – a) – \log_2 b)\]
Answer: \(5 \log_2 (1 – a) – 5 \log_2 b\)
4. Find the value of \(x\):
(i) \(\log 2 + \log x = 1\)
\[\log (2x) = 1\]
\[ 2x = 10^1 = 10\]
\[ x = \frac{10}{2} = 5\]
Answer: \(5\)
(ii) \(\log_2 x + \log_2 8 = 5\)
\[\log_2 (x \cdot 8) = 5\]
\[ x \cdot 8 = 2^5 = 32\]
\[ x = \frac{32}{8} = 4\]
Answer: \(4\)
(iii) \((81)^x = (243)^{x + 2}\)
Rewriting in base \(3\):
\[(3^4)^x = (3^5)^{x+2} \]
\[3^{4x} = 3^{5x+10}\]
\[4x = 5x + 10 \]
\[ x = -10\]
Answer: \(-10\)
(iv) \(\left(\frac{1}{27}\right)^{x-6} = 27\)
Rewriting in base \(3\):
\[\left(3^{-3}\right)^{x-6} = 3^3\]
\[ 3^{-3(x-6)} = 3^3\]
\[-3(x-6) = 3 \]
\[ -3x + 18 = 3 \]
\[ -3x = -15\]
\[ x = 5\]
Answer: \(5\)
(v) \( \log(5x – 10) = 2 \)
\[\log(5x – 10) = 2\]
\[5x – 10 = 10^2\]
\[5x – 10 = 100\]
\[5x = 110\]
\[x = 22\]
(vi) \( \log_2(x + 1) – \log_2(x – 4) = 2 \)
\[\log_2\left(\frac{x + 1}{x – 4}\right) = 2\]
\[\frac{x + 1}{x – 4} = 2^2\]
\[\frac{x + 1}{x – 4} = 4\]
\[x + 1 = 4(x – 4)\]
\[x + 1 = 4x – 16\]
\[1 + 16 = 4x – x\]
\[17 = 3x\]
\[x = \frac{17}{3}\]
Question 5: Solve the following using logarithm table:
(i) \(\frac{3.68 \times 4.21}{5.234}\)
Using logarithmic properties:
\[\log \left(\frac{3.68 \times 4.21}{5.234}\right) = \log(3.68) + \log(4.21) – \log(5.234)\]
From logarithm table:
\[\log(3.68) = 0.5658, \quad \log(4.21) = 0.6242, \quad \log(5.234) = 0.7190\]
\[\log(\text{Result}) = 0.5658 + 0.6242 – 0.7190 = 0.4710\]
\[\text{Result} = 10^{0.4710} = 2.95\]
(ii) \(4.67 \times 2.11 \times 2.397\)
Using logarithmic properties:
\[\log(4.67 \times 2.11 \times 2.397) = \log(4.67) + \log(2.11) + \log(2.397)\]
From logarithm table:
\[\log(4.67) = 0.6690, \quad \log(2.11) = 0.3243, \quad \log(2.397) = 0.3798\]
\[\log(\text{Result}) = 0.6690 + 0.3243 + 0.3798 = 1.3731\]
\[\text{Result} = 10^{1.3731} = 23.62\]
(iii) \(\frac{(20.46)^2 \times 2.4122}{754.3}\)
Using logarithmic properties:
\[\log \left(\frac{(20.46)^2 \times 2.4122}{754.3}\right) = 2 \cdot \log(20.46) + \log(2.4122) – \log(754.3)\]
From logarithm table:
\[\log(20.46) = 1.3100, \quad \log(2.4122) = 0.3827, \quad \log(754.3) = 2.8772\]
\[\log(\text{Result}) = 2 \cdot 1.3100 + 0.3827 – 2.8772 = 0.1249\]
\[\text{Result} = 10^{0.1249} = 1.33\]
(iv) \(\frac{\sqrt[3]{9.364} \times 21.64}{3.21}\)
Using logarithmic properties:
\[\log \left(\frac{\sqrt[3]{9.364} \times 21.64}{3.21}\right) = \frac{1}{3} \cdot \log(9.364) + \log(21.64) – \log(3.21)\]
From logarithm table:
\[\log(9.364) = 0.9715, \quad \log(21.64) = 1.3351, \quad \log(3.21) = 0.5065\]
\[\log(\text{Result}) = \frac{1}{3} \cdot 0.9715 + 1.3351 – 0.5065 = 1.1335\]
\[\text{Result} = 10^{1.1335} = 13.60\]
Question 6: Magnitude of Earthquake
The formula to measure the magnitude of earthquakes is:
\[M = \log_{10} \left(\frac{A}{A_0}\right)\]
If \(A = 10,000\) and \(A_0 = 10\), find \(M\).
Solution:
Substitute values:
\[M = \log_{10} \left(\frac{10,000}{10}\right) = \log_{10}(1,000)\]
\[M = \log_{10}(10^3) = 3\]
\[M = 3\]
Question 7: Investment Doubling
Abdullah invested Rs. 100,000 in a savings scheme gaining interest at 5% per annum. The total value after \(t\) years is modelled as:
\[y = 100,000 \cdot (1.05)^t\]
Find \(t\) when the investment doubles (\(y = 200,000\)).
Solution:
Set \(y = 200,000\):
\[2 = (1.05)^t\]
Take \(\log\) on both sides:
\[\log(2) = t \cdot \log(1.05)\]
From logarithm table:
\[\log(2) = 0.3010, \quad \log(1.05) = 0.0212\]
\[t = \frac{0.3010}{0.0212} = 14.20 \, \text{years}\]
Question 8: Temperature at Altitude
The temperature decreases by 3% for every 100 meters gained in altitude. The formula is:
\[T = T_i \cdot 0.97^{\frac{h}{100}}\]
Find \(T\) at an altitude \(h = 500\) meters, where \(T_i = 20^\circ C\).
Solution:
Substitute \(T_i = 20\) and \(h = 500\):
\[T = 20 \cdot 0.97^{\frac{500}{100}} = 20 \cdot 0.97^5\]
Calculate \(0.97^5\):
\[\log(0.97) = -0.0130 \quad \Rightarrow \quad 5 \cdot \log(0.97) = -0.0650\]
\[10^{-0.0650} = 0.87\]
\[T = 20 \cdot 0.87 = 17.4^\circ C\]
