Problem 76 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. $$3 \log _{3} x+4 \log _{3} y-4 \log _{3} z$$

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \( log_3 ((x^3 \cdot y^4) / z^4) \)

Step 1: Reordering and Applying Power Rule

Rewrite the expression as \( log_3 (x^3) + log_3(y^4) - log_3(z^4) \). Here, the power rule \( log_b m^n = n \cdot log_b(m) \) has been applied.

Step 2: Apply the Multiplication and Division properties of Logarithms

Now combine the logarithmic expressions using the properties of multiplication and division into a single logarithm, we get \( log_3 ((x^3 \cdot y^4) / z^4) \). This step uses \( log_b(mn) = log_b(m) + log_b(n) \) and \( log_b(m/n) = log_b(m) - log_b(n) \).

Step 3: Final simplified form

Hence the simplified form of the given logarithmic expression is \( log_3 ((x^3 \cdot y^4) / z^4) \)

Key Concepts

Logarithm Power Rule

The Logarithm Power Rule is an essential tool when working with logarithmic expressions, especially when you need to condense them. This rule states that if you have a logarithm such as \( \log_b(m^n) \), you can rewrite it as \( n \cdot \log_b(m) \). Simply put, the exponent of the argument becomes a coefficient.This is particularly useful for simplifying expressions where variables are raised to a power within a log. Let’s see this rule in action:

• For the expression \( 3 \log_3 x \), it transforms into \( \log_3(x^3) \).


• In \( 4 \log_3 y \), it changes to \( \log_3(y^4) \).


• Lastly, \( -4 \log_3 z \) becomes \( \log_3(z^4) \) while keeping the negative sign.

Mastering this rule allows you to manipulate and condense logarithms confidently, preparing you for using the next properties, which include multiplication and division rules.

Multiplication Property of Logarithms

The Multiplication Property of Logarithms helps in combining logs that have addition in between them. The property is simple: if you have \( \log_b(m) + \log_b(n) \), it can be simplified to \( \log_b(mn) \). This means you can multiply the arguments of the logs and use just one logarithm.Here’s why this is beneficial:

• Combining \( \log_3(x^3) + \log_3(y^4) \) using this property gives \( \log_3(x^3 \cdot y^4) \).


• It not only simplifies expressions but also makes them more manageable, especially when the ultimate goal is condensing.


• Ensure the logarithms have the same base before using this property.

Understanding this property allows you to see "addition" inside logarithmic expressions as an opportunity to condense into a single logarithm, setting up for any division property that might follow.

Division Property of Logarithms

The Division Property of Logarithms is equally important as it allows you to tackle subtraction in logarithmic expressions. According to this property, \( \log_b(m) - \log_b(n) \) simplifies to \( \log_b(\frac{m}{n}) \). This involves dividing the arguments of the logarithms.In practical usage:

• This property was used to condense \( \log_3(x^3 \cdot y^4) - \log_3(z^4) \) into \( \log_3\left(\frac{x^3 \cdot y^4}{z^4}\right) \).


• Handle subtraction by thinking of division - separate logs with subtraction can lead to a "division" within a single log.


• Again, ensure all logs have the same base to correctly apply this rule.

With both multiplication and division properties, you can transform complex multi-logarithmic expressions into singular, elegant logs.