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Math  /  Algebra

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

Studdy Solution

STEP 1

1. We will use properties of logarithms to simplify and evaluate the expressions.
2. We assume familiarity with basic logarithmic identities and properties.

STEP 2

1. Evaluate log⁡3(127)\log _{3}\left(\frac{1}{27}\right).
2. Evaluate log⁡6(6)\log _{6}(\sqrt{6}).
3. Evaluate log⁡2(18)+4\log _{2}\left(\frac{1}{8}\right) + 4.
4. Evaluate 6log⁡8(4)6 \log _{8}(4).

STEP 3

To evaluate log⁡3(127)\log _{3}\left(\frac{1}{27}\right), recognize that 27=3327 = 3^3, so 127=3−3\frac{1}{27} = 3^{-3}.
Using the property of logarithms log⁡b(ba)=a\log_b(b^a) = a, we have:
log⁡3(127)=log⁡3(3−3)=−3\log _{3}\left(\frac{1}{27}\right) = \log _{3}(3^{-3}) = -3

STEP 4

To evaluate log⁡6(6)\log _{6}(\sqrt{6}), recognize that 6=61/2\sqrt{6} = 6^{1/2}.
Using the property of logarithms log⁡b(ba)=a\log_b(b^a) = a, we have:
log⁡6(6)=log⁡6(61/2)=12\log _{6}(\sqrt{6}) = \log _{6}(6^{1/2}) = \frac{1}{2}

STEP 5

To evaluate log⁡2(18)+4\log _{2}\left(\frac{1}{8}\right) + 4, recognize that 8=238 = 2^3, so 18=2−3\frac{1}{8} = 2^{-3}.
Using the property of logarithms log⁡b(ba)=a\log_b(b^a) = a, we have:
log⁡2(18)=log⁡2(2−3)=−3\log _{2}\left(\frac{1}{8}\right) = \log _{2}(2^{-3}) = -3
Then, add 4:
−3+4=1-3 + 4 = 1

STEP 6

To evaluate 6log⁡8(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 log⁡b(ba)=a\log_b(b^a) = a, we have:
log⁡8(4)=log⁡8(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. log⁡3(127)=−3\log _{3}\left(\frac{1}{27}\right) = -3
2. log⁡6(6)=12\log _{6}(\sqrt{6}) = \frac{1}{2}
3. log⁡2(18)+4=1\log _{2}\left(\frac{1}{8}\right) + 4 = 1
4. 6log⁡8(4)=26 \log _{8}(4) = 2

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