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Math  /  Algebra

QuestionWrite the expression as a sum and/or difference of logarithms. Express exponents as factors. log⁡6xy3\log _{6} \frac{\sqrt{x}}{y^{3}} A) 32(log⁡6x−log⁡6y)\frac{3}{2}\left(\log _{6} x-\log _{6} y\right) B) log⁡6x2−log⁡63y\log _{6} \frac{x}{2}-\log _{6} 3 y C) 12log⁡6x+3log⁡6y\frac{1}{2} \log _{6} x+3 \log _{6} y D) 12log⁡6x−3log⁡6y\frac{1}{2} \log _{6} x-3 \log _{6} y

Studdy Solution

STEP 1

1. We are given a logarithmic expression and need to simplify it.
2. The properties of logarithms, such as the quotient rule and the power rule, can be applied to simplify the expression.

STEP 2

1. Apply the quotient rule of logarithms.
2. Apply the power rule of logarithms to express exponents as factors.

STEP 3

Apply the quotient rule of logarithms, which states that:
log⁡b(MN)=log⁡bM−log⁡bN\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N
For the given expression:
log⁡6xy3=log⁡6x−log⁡6y3\log_6 \frac{\sqrt{x}}{y^3} = \log_6 \sqrt{x} - \log_6 y^3

STEP 4

Apply the power rule of logarithms, which states that:
log⁡b(Mn)=n⋅log⁡bM\log_b (M^n) = n \cdot \log_b M
For the expression log⁡6x\log_6 \sqrt{x}, recognize that x=x1/2\sqrt{x} = x^{1/2}:
log⁡6x=log⁡6(x1/2)=12log⁡6x\log_6 \sqrt{x} = \log_6 (x^{1/2}) = \frac{1}{2} \log_6 x
For the expression log⁡6y3\log_6 y^3:
log⁡6y3=3⋅log⁡6y\log_6 y^3 = 3 \cdot \log_6 y

STEP 5

Combine the results from the power rule:
log⁡6xy3=12log⁡6x−3log⁡6y\log_6 \frac{\sqrt{x}}{y^3} = \frac{1}{2} \log_6 x - 3 \log_6 y
This matches option D.
The correct answer is:
D \boxed{D}

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