Example Question - simplifying logarithmic expressions

Simplifying Logarithmic Expressions

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.