Logarithmic Expressions: Difficult Problems with Solutions

Solution:
Using [tex]\log_a x - \log_a y = \log_a\frac{x}{y}[/tex]:
[tex]\log_{2}6 - \log_{2}3 = \log_{2}\frac{6}{3} = \log_{2}2 = 1[/tex]

Answer: 1

Solution:
Using [tex]\log_a x + \log_a y = \log_a(xy)[/tex]:
[tex]\log_{3}5 + \log_{3}1.8 = \log_{3}(5\cdot 1.8) = \log_{3}9 = 2[/tex]

Answer: 2

Solution:
Using [tex]\log_a x + \log_a y = \log_a(xy)[/tex]:
[tex]\log_{10}2 + \log_{10}5 = \log_{10}10 = 1[/tex]

Answer: 1

Solution:
[tex]\frac{1}{2} = 2^{-1}[/tex]
[tex]8 = 2^3[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{\frac{1}{2}}8 = \log_{2^{-1}}2^3 = \frac{3}{-1} = -3[/tex]

Answer: -3

Solution:
Using [tex]a^{\log_a b} = b[/tex]:
[tex]5^{\log_{5}7} = 7[/tex]

Answer: 7

Solution:
[tex]4 = 2^2[/tex]
[tex]8 = 2^3[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{4}8 = \log_{2^2}2^3 = \frac{3}{2}[/tex]

Answer: [tex]\frac{3}{2}[/tex]

Solution:
Using [tex]\log_a b\cdot \log_b c = \log_a c[/tex]:
[tex]\log_{2}3\cdot \log_{3}8 = \log_{2}8 = 3[/tex]

Answer: 3

Solution:
Using [tex]\log_a x + \log_a y = \log_a(xy)[/tex]:
[tex]\log_{6}2 + \log_{6}18 = \log_{6}36 = 2[/tex]

Answer: 2

Solution:
Using [tex]\log_a x - \log_a y = \log_a\frac{x}{y}[/tex]:
[tex]\log_{2}80 - \log_{2}5 = \log_{2}16 = 4[/tex]

Answer: 4

Solution:
[tex]\sqrt{3} = 3^{\frac{1}{2}}[/tex]
[tex]9 = 3^2[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{\sqrt{3}}9 = \log_{3^{\frac{1}{2}}}3^2 = \frac{2}{\frac{1}{2}} = 4[/tex]

Answer: 4

Solution:
[tex]3^{2\log_{3}5} = (3^{\log_{3}5})^2[/tex]
Using [tex]a^{\log_a b} = b[/tex]:
[tex](3^{\log_{3}5})^2 = 5^2 = 25[/tex]

Answer: 25

Solution:
[tex]\log_{2}16 = \log_{2}2^4 = 4[/tex]
[tex]\log_{2}(\log_{2}16) = \log_{2}4 = 2[/tex]

Answer: 2

Solution:
[tex]49 = 7^2[/tex]
[tex]49^3 = (7^2)^3 = 7^6[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{7}49^3 = \log_{7}7^6 = 6[/tex]

Answer: 6

Solution:
[tex]\log_{10}7 + \log_{10}2 = \log_{10}(7\cdot 2) = \log_{10}14[/tex]
Using [tex]a^{\log_a b} = b[/tex]:
[tex]10^{\log_{10}14} = 14[/tex]

Answer: 14

Solution:
[tex]\frac{1}{\sqrt{5}} = \frac{1}{5^{\frac{1}{2}}} = 5^{-\frac{1}{2}}[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{5}\frac{1}{\sqrt{5}} = \log_{5}5^{-\frac{1}{2}} = -\frac{1}{2}[/tex]

Answer: [tex]-\frac{1}{2}[/tex]

Solution:
[tex]2\sqrt[3]{2} = 2\cdot 2^{\frac{1}{3}} = 2^{1+\frac{1}{3}} = 2^{\frac{4}{3}}[/tex]
[tex]\frac{1}{16} = \frac{1}{2^4} = 2^{-4}[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{2\sqrt[3]{2}}\frac{1}{16} = \log_{2^{\frac{4}{3}}}2^{-4} = \frac{-4}{\frac{4}{3}} = -4\cdot\frac{3}{4} = -3[/tex]

Answer: -3

Solution:
[tex]\frac{1}{8} = \frac{1}{2^3} = 2^{-3}[/tex]
[tex]\sqrt[4]{32} = \sqrt[4]{2^5} = 2^{\frac{5}{4}}[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{\frac{1}{8}}\sqrt[4]{32} = \log_{2^{-3}}2^{\frac{5}{4}} = \frac{\frac{5}{4}}{-3} = \frac{5}{4}\cdot\left(-\frac{1}{3}\right) = -\frac{5}{12}[/tex]

Answer: [tex]-\frac{5}{12}[/tex]

Solution:
[tex]\frac{1}{8} = \frac{1}{2^3} = 2^{-3}[/tex]
[tex]0.25 = \frac{1}{4} = \frac{1}{2^2} = 2^{-2}[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{\frac{1}{8}}0.25 = \log_{2^{-3}}2^{-2} = \frac{-2}{-3} = \frac{2}{3}[/tex]

Answer: [tex]\frac{2}{3}[/tex]