Problem 46 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\frac{3}{2} \log 16 x^{4}-\frac{1}{2} \log y^{8}$$

Step-by-Step Solution

Answer

The given logarithmic expression simplifies to \(2 \log(4096/y^2)\)

Step 1: Apply the power rule of logarithms

We can firstly apply the power rule \(log(a^{n}) = n \cdot log(a)\) to the expression. Applying this rule, the given expression becomes: \(\frac{3}{2}(4 log 16) - \frac{1}{2}(8 log y) = 6 \log 16 - 4 log y\)

Step 2: Apply the quotient rule of logarithms

Now we can apply the quotient rule \(log(a) - log(b) = log(a/b)\) to further simplify: \(6 \log 16 - 4 log y = 2(3 log 16 - 2 log y) = 2 log(16^3/y^2) = 2 log(4096/y^2)\)

Step 3: Final Simplification

The final step would be simplifying the expression, however, without any other constraints mentioned about the values of y, this is the final simple form that can be achieved.

Key Concepts

Power Rule of Logarithms

Understanding the power rule of logarithms is crucial when simplifying logarithmic expressions. This rule is especially handy when you're faced with exponents in a logarithmic function. To put it simply, the power rule states that \(\log(a^{n}) = n \cdot \log(a)\), where \(a\) is the base of the log, and \(n\) is the exponent.

For instance, imagine you have \(\log(16x^{4})\). Applying the power rule, you can transform it to \(4\log(16) + \log(x)\), which is more approachable. In our exercise, the term \(\frac{3}{2} \log(16x^4)\) becomes \(6 \log(16)\) after multiplying by the coefficient. The power rule turns the intimidating exponents into friendlier coefficients that multiply the log of the base, simplifying the expression significantly.

Quotient Rule of Logarithms

The quotient rule of logarithms is another tool that simplifies log expressions involving division. If you have the expression \(\log(A) - \log(B)\), the quotient rule allows you to combine these two logs into a single one, showing the division of \(A\) and \(B\). Mathematically, it's written as \(\log(A) - \log(B) = \log(\frac{A}{B})\).

Applying this rule to our exercise, we have \(6 \log(16) - 4 \log(y)\). The coefficient of 4 can be incorporated back into the log as an exponent, leading to \(\log(16^3) - \log(y^2)\), and further simplifying to \(\log(\frac{16^3}{y^2})\). Combining terms using the quotient rule not only tidies up an equation, but it also sets the stage for any further simplification if possible.

Logarithmic Expressions

Logarithmic expressions are representations that involve logs, commonly in the form of \(\log_{b}(x)\) which asks 'to what power do we raise \(b\) to get \(x\)?'. Simplifying these expressions is a matter of recognizing patterns and applying rules like the power and quotient rules we've discussed. Doing so often simplifies the log to a more digestible form.

However, it's important to note that simplification does not always result in a numerical value. As seen in our final simplified form \(2 \log(4096/y^2)\), we can't progress further without additional information on \(y\). Nevertheless, we've achieved a single logarithmic statement rather than an initially more complex expression, demonstrating the utility of these rules in reducing complexity.