Math  /  Algebra

QuestionNumeric For the following exercises, evaluate the base bb logarithmic expression without using a calculator.
42. log3(127)\log _{3}\left(\frac{1}{27}\right)
43. log6(6)\log _{6}(\sqrt{6})
44. log2(18)+4\log _{2}\left(\frac{1}{8}\right)+4
45. 6log8(4)6 \log _{8}(4)

Studdy Solution
To evaluate 6log8(4)6 \log _{8}(4), recognize that 4=81/34 = 8^{1/3} because 4=224 = 2^2 and 8=238 = 2^3, so 81/3=22/38^{1/3} = 2^{2/3}.
Using the property of logarithms logb(ba)=a\log_b(b^a) = a, we have:
log8(4)=log8(81/3)=13\log _{8}(4) = \log _{8}(8^{1/3}) = \frac{1}{3}
Then, multiply by 6:
6×13=26 \times \frac{1}{3} = 2
The evaluated expressions are:
1. log3(127)=3\log _{3}\left(\frac{1}{27}\right) = -3
2. log6(6)=12\log _{6}(\sqrt{6}) = \frac{1}{2}
3. log2(18)+4=1\log _{2}\left(\frac{1}{8}\right) + 4 = 1
4. 6log8(4)=26 \log _{8}(4) = 2

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