Logarithmic Expressions: Problems with Solutions

Solution:
[tex]16 = 2^4[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{2}16 = \log_{2}2^4 = 4[/tex]

Answer: 4

Solution:
[tex]81 = 3^4[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{3}81 = \log_{3}3^4 = 4[/tex]

Answer: 4

Solution:
[tex]125 = 5^3[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{5}125 = \log_{5}5^3 = 3[/tex]

Answer: 3

Solution:
[tex]0.01 = \frac{1}{100} = \frac{1}{10^2} = 10^{-2}[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{10}0.01 = \log_{10}10^{-2} = -2[/tex]

Answer: -2

Solution:
[tex]\frac{1}{32} = \frac{1}{2^5} = 2^{-5}[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{2}\frac{1}{32} = \log_{2}2^{-5} = -5[/tex]

Answer: -5

Solution:
[tex]1 = 7^0[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{7}1 = \log_{7}7^0 = 0[/tex]

Answer: 0

Solution:
[tex]6 = 6^1[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{6}6 = \log_{6}6^1 = 1[/tex]

Answer: 1

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

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

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

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

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

Answer: [tex]\frac{1}{3}[/tex]

Solution:
[tex]25 = 5^2[/tex]
[tex]125 = 5^3[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{25}125 = \log_{5^2}5^3 = \frac{3}{2}[/tex]

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

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

Answer: 3

Solution:
[tex]\frac{1}{3} = 3^{-1}[/tex]
[tex]27 = 3^3[/tex]
Using [tex]\log_{a^n}a^m = \frac{m}{n}[/tex]:
[tex]\log_{\frac{1}{3}}27 = \log_{3^{-1}}3^{3} = \frac{3}{-1} = -3[/tex]

Answer: -3

Solution:
[tex]256 = 4^4[/tex]
[tex]\frac{1}{256} = \frac{1}{4^4} = 4^{-4}[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_4(\frac{1}{256}) = \log_4 4^{-4} = -4[/tex]

Answer: -4

Solution:
ln is a logarithm whose base is e, then [tex]ln e^{3} = 3ln e = 3 [/tex]

Solution:
[tex]0.001 = \frac{1}{1000} = 10^{-3}[/tex]
Using [tex]\log_a a^m = m[/tex]:
[tex]\log_{10}0.001 = \log_{10}10^{-3} = -3[/tex]

Answer: -3

Solution:
Because the bases are greater than 1, the logarithmic functions are increasing.
[tex]\log_{3}6 < \log_{3}8 [/tex]

Solution:
Because the bases are smaller than 1, the logarithmic function is decreasing.
[tex]\log_{\frac{1}{3}}6 > \log_{\frac{1}{3}}8 [/tex]

Solution:
By the definition of a logarithm, [tex]\log_{2}3[/tex] is the exponent to which 2 must be raised to obtain 3, therefore:
[tex]2^{\log_{2}3} = 3[/tex]

In general [tex]a^{\log_a b} = b[/tex], [tex]a>0, a\neq 1, b>0[/tex].

Answer: 3

Solution:
[tex]\log_{3}3^{15}=15\log_{3}3= 15 [/tex]

Solution:
[tex]\log^2_{3}3^{5}=25\log^2_{3}3 = 25 [/tex]

Solution:
[tex]\log_{5}8 + \log_{5}7 = \log_{5}56 [/tex]

Solution:
[tex]\log_{3}42 - \log_{3}7 =\log_{3}{\frac{42}{7}}=\log_{3}6 [/tex]

Solution:
[tex] \log_{3}8 = \frac {1}{\log_{8}3} [/tex]

Solution:
[tex]\log_{9}{7}={\frac{\log_{3}7}{\log_{3}9}}={\frac{\log_{3}7}{2}} [/tex]

Solution:
[tex]2\log_{4}6 - \log_{4} 9 = \log_{4}6^{2}-\log_{4} 9 =\log_{4}36-\log_{4}9=\log_{4} 4 =1 [/tex]

Solution:
[tex]\log_{3}10 + \log_{9} 20 = \log_{3} 10 + {\frac{\log_{3}20}{\log_{3}9}}=\log_{3}10+{\frac{\log_{3}20}{2}}=\log_{3}10 + \log_{3}{\sqrt{20} }=\log_{3}10\sqrt{20}[/tex]