Problem 36 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

Step-by-Step Solution

Answer

\((1/5)*[\log_{2}(x) + 4*\log_{2}(y) - 4]\)

Step 1: Convert Fractional Exponent into Radical Form

Let's start by writing the fractional exponent inside the logarithm as a radical: \n\n\(\log_{2} ((x y^{4}/16)^{1/5})\)

Step 2: Use the Logarithmic Power Rule

Now use the logarithmic power rule (log(b^p)=p log(b)) to bring the exponent in front of the logarithm: \n\n\((1/5)*\log_{2}(x y^{4}/16)\)

Step 3: Express Logarithm of Quotient

Next, use the logarithm product rule \(\log(ab) = \log(a) + \log(b)\) and quotient rule \(\log(a/b) = \log(a) - \log(b)\) to express this as subtraction of logarithms: \n\n\((1/5)*[\log_{2}(x) + \log_{2}(y^{4}) - \log_{2}(16)]\)

Step 4: Apply Logarithmic Power Rule

We're not done yet. Let's apply the logarithmic power rule again to bring the 4 in front of one of the logarithms: \n\n\((1/5)*[\log_{2}(x) + 4*\log_{2}(y) - \log_{2}(16)]\)

Step 5: Simplify

Finally, let's simplify the logarithm base 2 of 16, because we know that \(2^{4} = 16\): \n\n\((1/5)*[\log_{2}(x) + 4*\log_{2}(y) - 4]\)