Exercise 4.2: Logarithms
1. Find the values:
1) $\log_3 81$
Solution: $\log_3 81$
$= \log_3 (3^4)$
$= 4 \log_3 3$$[\because \log_a (M^r) = r \log_a M]$
$= 4 \times 1$$[\because \log_a a = 1]$
$= 4$
Answer:$4$
2) $\log_5 \sqrt[3]{5}$
Solution: $\log_5 \sqrt[3]{5}$
$= \log_5 (5^{\frac{1}{3}})$
$= \frac{1}{3} \log_5 5$
$= \frac{1}{3} \times 1 = \frac{1}{3}$
Answer:$\frac{1}{3}$
3) $\log_4 2$
Solution: $\log_4 2$
$= \log_4 (\sqrt{4})$
$= \log_4 (4^{\frac{1}{2}})$
$= \frac{1}{2} \log_4 4$
$= \frac{1}{2} \times 1 = \frac{1}{2}$
Answer:$\frac{1}{2}$
4) $\log_{2\sqrt{5}} 400$
Solution: $\log_{2\sqrt{5}} 400$
$= \log_{2\sqrt{5}} (2\sqrt{5})^4$$[\because (2\sqrt{5})^4 = 2^4 \times (\sqrt{5})^4 = 16 \times 25 = 400]$
$= 4 \log_{2\sqrt{5}} (2\sqrt{5})$
$= 4 \times 1 = 4$
Answer:$4$
5) $\log_5 (\sqrt[3]{5} \cdot \sqrt{5})$
Solution: $\log_5 (\sqrt[3]{5} \cdot \sqrt{5})$
$= \log_5 (5^{\frac{1}{3}} \cdot 5^{\frac{1}{2}})$
$= \log_5 (5^{\frac{1}{3} + \frac{1}{2}})$
$= \log_5 (5^{\frac{2+3}{6}})$
$= \log_5 (5^{\frac{5}{6}})$
$= \frac{5}{6} \log_5 5$
$= \frac{5}{6} \times 1 = \frac{5}{6}$
Answer:$\frac{5}{6}$
2. Find the value of $x$:
1) $\log_5 x = 3$
Solution: $\log_5 x = 3$
or, $x = 5^3$$[\because \log_a N = x \iff a^x = N]$
or, $x = 125$
Answer:$125$
2) $\log_x 25 = 2$
Solution: $\log_x 25 = 2$
or, $x^2 = 25$
or, $x^2 = 5^2$
or, $x = 5$$[x > 0]$
Answer:$5$
3) $\log_x \frac{1}{16} = -2$
Solution: $\log_x \frac{1}{16} = -2$
or, $x^{-2} = \frac{1}{16}$
or, $\frac{1}{x^2} = \frac{1}{16}$
or, $x^2 = 16$
or, $x = \sqrt{16}$
or, $x = 4$
Answer:$4$
3. Show that:
1) $5\log_{10}5 – \log_{10}25 = \log_{10}125$
Solution:
L.H.S. $= 5\log_{10}5 – \log_{10}25$
$= \log_{10}(5^5) – \log_{10}(5^2)$
$= \log_{10}\left(\frac{5^5}{5^2}\right)$
$= \log_{10}(5^{5-2})$
$= \log_{10}(5^3)$
$= \log_{10}125 = \text{R.H.S.}$
$\therefore 5\log_{10}5 – \log_{10}25 = \log_{10}125$ (Shown)
2) $\log_{10}\frac{50}{147} = \log_{10}2 + 2\log_{10}5 – \log_{10}3 – 2\log_{10}7$
Solution:
L.H.S. $= \log_{10}\frac{50}{147}$
$= \log_{10}50 – \log_{10}147$
$= \log_{10}(2 \times 5^2) – \log_{10}(3 \times 7^2)$
$= (\log_{10}2 + \log_{10}5^2) – (\log_{10}3 + \log_{10}7^2)$
$= \log_{10}2 + 2\log_{10}5 – \log_{10}3 – 2\log_{10}7 = \text{R.H.S.}$
$\therefore \log_{10}\frac{50}{147} = \log_{10}2 + 2\log_{10}5 – \log_{10}3 – 2\log_{10}7$ (Shown)
3) $3\log_{10}2 + 2\log_{10}3 + \log_{10}5 = \log_{10}360$
Solution:
L.H.S. $= 3\log_{10}2 + 2\log_{10}3 + \log_{10}5$
$= \log_{10}(2^3) + \log_{10}(3^2) + \log_{10}5$
$= \log_{10}8 + \log_{10}9 + \log_{10}5$
$= \log_{10}(8 \times 9 \times 5)$
$= \log_{10}360 = \text{R.H.S.}$
$\therefore 3\log_{10}2 + 2\log_{10}3 + \log_{10}5 = \log_{10}360$ (Shown)
4. Simplify:
1) $7\log_{10}\frac{10}{9} – 2\log_{10}\frac{25}{24} + 3\log_{10}\frac{81}{80}$
Solution: $7\log_{10}\frac{10}{9} – 2\log_{10}\frac{25}{24} + 3\log_{10}\frac{81}{80}$
$= \log_{10}\left(\frac{10}{9}\right)^7 – \log_{10}\left(\frac{25}{24}\right)^2 + \log_{10}\left(\frac{81}{80}\right)^3$
$= \log_{10}\left(\frac{2 \times 5}{3^2}\right)^7 – \log_{10}\left(\frac{5^2}{2^3 \times 3}\right)^2 + \log_{10}\left(\frac{3^4}{2^4 \times 5}\right)^3$
$= \log_{10}\left( \frac{2^7 \times 5^7}{3^{14}} \div \frac{5^4}{2^6 \times 3^2} \times \frac{3^{12}}{2^{12} \times 5^3} \right)$
$= \log_{10}\left( \frac{2^7 \times 5^7}{3^{14}} \times \frac{2^6 \times 3^2}{5^4} \times \frac{3^{12}}{2^{12} \times 5^3} \right)$
$= \log_{10}\left( \frac{2^{7+6} \times 3^{2+12} \times 5^7}{2^{12} \times 3^{14} \times 5^{4+3}} \right)$
$= \log_{10}\left( \frac{2^{13} \times 3^{14} \times 5^7}{2^{12} \times 3^{14} \times 5^7} \right)$
$= \log_{10}(2^{13-12})$
$= \log_{10}2$
Answer:$\log_{10}2$
2) $\log_7(\sqrt[5]{7} \cdot \sqrt{7}) – \log_3\sqrt[3]{3} + \log_4 2$
Solution: $\log_7(\sqrt[5]{7} \cdot \sqrt{7}) – \log_3\sqrt[3]{3} + \log_4 2$
$= \log_7(7^{\frac{1}{5}} \cdot 7^{\frac{1}{2}}) – \log_3(3^{\frac{1}{3}}) + \log_4(\sqrt{4})$
$= \log_7(7^{\frac{1}{5} + \frac{1}{2}}) – \frac{1}{3}\log_3 3 + \log_4(4^{\frac{1}{2}})$
$= \log_7(7^{\frac{7}{10}}) – \frac{1}{3}(1) + \frac{1}{2}\log_4 4$
$= \frac{7}{10}\log_7 7 – \frac{1}{3} + \frac{1}{2}(1)$
$= \frac{7}{10} – \frac{1}{3} + \frac{1}{2}$
$= \frac{21 – 10 + 15}{30}$
$= \frac{26}{30} = \frac{13}{15}$
Answer:$\frac{13}{15}$
3) $\log_e\frac{a^3 b^3}{c^3} + \log_e\frac{b^3 c^3}{d^3} + \log_e\frac{c^3 d^3}{a^3} – 3\log_e b^2 c$
Solution: $\log_e\frac{a^3 b^3}{c^3} + \log_e\frac{b^3 c^3}{d^3} + \log_e\frac{c^3 d^3}{a^3} – 3\log_e b^2 c$
$= \log_e\left( \frac{a^3 b^3}{c^3} \times \frac{b^3 c^3}{d^3} \times \frac{c^3 d^3}{a^3} \right) – \log_e (b^2 c)^3$
$= \log_e(b^6 c^3) – \log_e(b^6 c^3)$
$= 0$
Answer:$0$
5. $x = 2, y = 3, z = 5, w = 7$
1) What is the log of $\sqrt{y^3}$ to the base $3$.
Solution:
When $y = 3$, $\sqrt{y^3} = \sqrt{3^3} = 3^{\frac{3}{2}}$
Therefore, we get
$log_3\sqrt{y^3}$
$= \log_3 (3)^\frac{3}{2} $
$= \frac{3}{2} \log_3 3 = \frac{3}{2} \times 1 = \frac{3}{2}$
Answer:$\frac{3}{2}$
2) Find the value of $w\log\frac{xz}{y^2} – x\log\frac{z^2}{x^2 y} + y\log\frac{y^4}{x^4 z}$.
Solution:
Given, $x=2, y=3, z=5, w=7$
Given expression $= w\log\frac{xz}{y^2} – x\log\frac{z^2}{x^2 y} + y\log\frac{y^4}{x^4 z}$
$= 7\log\frac{2 \times 5}{3^2} – 2\log\frac{5^2}{2^2 \times 3} + 3\log\frac{3^4}{2^4 \times 5}$
$= \log_{10}\left(\frac{10}{9}\right)^7 – \log_{10}\left(\frac{25}{24}\right)^2 + \log_{10}\left(\frac{81}{80}\right)^3$
$= \log_{10}\left(\frac{2 \times 5}{3^2}\right)^7 – \log_{10}\left(\frac{5^2}{2^3 \times 3}\right)^2 + \log_{10}\left(\frac{3^4}{2^4 \times 5}\right)^3$
$= \log_{10}\left( \frac{2^7 \times 5^7}{3^{14}} \div \frac{5^4}{2^6 \times 3^2} \times \frac{3^{12}}{2^{12} \times 5^3} \right)$
$= \log_{10}\left( \frac{2^7 \times 5^7}{3^{14}} \times \frac{2^6 \times 3^2}{5^4} \times \frac{3^{12}}{2^{12} \times 5^3} \right)$
$= \log_{10}\left( \frac{2^{7+6} \times 3^{2+12} \times 5^7}{2^{12} \times 3^{14} \times 5^{4+3}} \right)$
$= \log_{10}\left( \frac{2^{13} \times 3^{14} \times 5^7}{2^{12} \times 3^{14} \times 5^7} \right)$
$= \log_{10}(2^{13-12})$
$= \log_{10}2$
Answer:$\log_{10}2$
3) Show that, $\frac{\log\sqrt{y^3} + y\log x – \frac{y}{x}\log(xz)}{\log(xy) – \log z} = \log_y\sqrt{y^3}$
Solution:
Substituting $x=2, y=3, z=5$:
R.H.S. $= \log_y\sqrt{y^3} = \log_3 (3^{\frac{3}{2}}) = \frac{3}{2} \log_3 3 = \frac{3}{2}$
L.H.S. $= \frac{\log\sqrt{3^3} + 3\log 2 – \frac{3}{2}\log(2 \times 5)}{\log(2 \times 3) – \log 5}$
$= \frac{\log (3^{\frac{3}{2}}) + \log (2^3) – \frac{3}{2}\log 10}{\log 6 – \log 5}$
$= \frac{\frac{3}{2}\log 3 + 3\log 2 – \frac{3}{2}(\log 2 + \log 5)}{\log\frac{6}{5}}$
$= \frac{\frac{3}{2}\log 3 + 3\log 2 – \frac{3}{2}\log 2 – \frac{3}{2}\log 5}{\log\frac{6}{5}}$
$= \frac{\frac{3}{2}\log 3 + \frac{3}{2}\log 2 – \frac{3}{2}\log 5}{\log\frac{6}{5}}$
$= \frac{\frac{3}{2}(\log 3 + \log 2 – \log 5)}{\log\left(\frac{2 \times 3}{5}\right)}$
$= \frac{\frac{3}{2}\log\left(\frac{6}{5}\right)}{\log\left(\frac{6}{5}\right)} = \frac{3}{2} = \text{R.H.S.}$
$\therefore \frac{\log\sqrt{y^3} + y\log x – \frac{y}{x}\log(xz)}{\log(xy) – \log z} = \log_y\sqrt{y^3}$ (Shown)