Write the following expression as a sum or difference of logarithms with no exponents. Assume all variables represent positive values Simplify your answer completely. log_(4)((y^16z^4)/(sqrt [2](w+15))) log_(4)((y^16z^4)/(sqrt [3](w+15)))=square Importance of Concept Like many operations in mathematics, the process of expanding a logarithm might seem abstract and not useful; however, many challenging problems in calculus (specifically, differential calculus can be greatly simplified by applying logarithms and expanding the resulting expression.

To write the given expression as a sum or difference of logarithms with no exponents, we can use the properties of logarithms. <br /><br />The given expression is:<br />\[ \log_{4}\left(\frac{y^{16}z^{4}}{\sqrt[2]{w+15}}\right) \]<br /><br />We can use the quotient rule of logarithms, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. <br /><br />So, we can rewrite the expression as:<br />\[ \log_{4}(y^{16}z^{4}) - \log_{4}(\sqrt[2]{w+15}) \]<br /><br />Next, we can use the product rule of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the factors. <br /><br />Applying the product rule to the first term, we get:<br />\[ \log_{4}(y^{16}) + \log_{4}(z^{4}) \]<br /><br />Now, we can simplify the expression further by using the power rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. <br /><br />Applying the power rule to the terms, we get:<br />\[ 16\log_{4}(y) + 4\log_{4}(z) - \frac{1}{2}\log_{4}(w+15) \]<br /><br />Therefore, the expanded form of the given expression is:<br />\[ \log_{4}\left(\frac{y^{16}z^{4}}{\sqrt[2]{w+15}}\right) = 16\log_{4}(y) + 4\log_{4}(z) - \frac{1}{2}\log_{4}(w+15) \]<br /><br />This expression is now written as a sum and difference of logarithms with no exponents.