Problem 11 - 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. $$ \log _{4}\left(\frac{64}{y}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(3 - \log _{4}(y)\)

Step 1: Apply the division rule

First, to expand the logarithm, let's use the fact that for any two positive real numbers \(a\) and \(b\), the logarithm division rule states that \(\log_b\left(\frac{a}{b}\right) = \log_b(a) - \log_b(b)\). So \(\log _{4}\left(\frac{64}{y}\right) = \log _{4}(64) - \log _{4}(y) \)

Step 2: Apply the exponent rule

Now apply the other rule that for any positive real number \(b\) and any integer \(n\), if \(64=b^n\), then \(\log_b(b^n) = n\). Hence, \(\log _{4} 64\) can be simplified as \(\log _{4} 4^3 = 3\). So the expression now will be \(3 - \log _{4}(y)\)

Step 3: Final Simplification

No further simplification can be done, so this is the final answer.

Key Concepts

Logarithm Division Rule

Understanding the logarithm division rule is vital when you're faced with a logarithmic expression involving division. Picture this rule as a way of breaking down a complex log into simpler pieces. If you have \( \log_b\left(\frac{a}{c}\right) \), this can be expanded to \( \log_b(a) - \log_b(c) \). Simply put, the log of a quotient is the difference of the logs.

For example, consider \( \log_{4}\left(\frac{64}{y}\right) \). Here, 64 is the numerator and y is the denominator. Applying the division rule gives us \( \log_{4}(64) - \log_{4}(y) \). It's like breaking a log in two, helping you to tackle each part separately and confidently.

Logarithmic Expression Simplification

The art of logarithmic expression simplification is akin to cleaning up a messy room, organizing different elements and leaving the space clear. It’s about combining the properties of logarithms to rewrite expressions in their simplest form—that includes using rules like the division rule we discussed earlier.

If you've got \( \log_{4}(64) - \log_{4}(y) \), clearly \(64 = 4^3\), which means we can transform \( \log_{4}(64) \) by using what's known as the exponent rule in logarithms. Simplification is like distilling the essence of the expression, and after applying the exponent rule, you’re left with a beautified version: \(3 - \log_{4}(y)\). It’s all about making the complex simple, and who doesn't love simplicity?

Exponent Rule in Logarithms

The Power of 'Power'

The exponent rule in logarithms gives you the superpower to handle logarithms of numbers raised to a power. Think about when you see \( \log_b(b^n) \), where \(b \) is the base of the logarithm and \(n \) is the exponent. The rule tells us that this is equal to \(n\), the exponent itself. It's like the base and the log cancel each other out, and only the exponent remains standing.

So, looking at our previous example where we simplified \( \log_{4}(64) \) to 3, it’s because we recognized that \(64 \) was really \(4^3\), and applying this rule, we knew that \( \log_{4}(4^3) = 3\). This exponent rule essentially serves as a bridge, guiding you from a complex log to a simple and elegant numerical value.