Question - Simplifying Logarithmic Expressions
Solution:
The expression provided in the image, $$\log\left(\frac{a^3b}{c^2}\right)$$, can be simplified using the properties of logarithms:The Quotient Rule: $$\log(x/y) = \log(x) - \log(y)$$The Power Rule: $$\log(x^k) = k \cdot \log(x)$$So let's break down the expression step by step:$$\log\left(\frac{a^3b}{c^2}\right)$$= $$ \log(a^3b) - \log(c^2) $$ (using the Quotient Rule)= $$ \log(a^3) + \log(b) - \log(c^2) $$ (using the Product Rule: $$\log(xy) = \log(x) + \log(y)$$)= $$ 3\cdot\log(a) + \log(b) - 2\cdot\log(c) $$ (using the Power Rule)Thus, the correct answer to the expression given in the image is:$$\log(a) + 3\log(b) - 2\log(c)$$This corresponds to answer choice D in the image.
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