Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\frac{1}{2} \log x-\frac{1}{3} \log y-2 \log z$$

Step-by-Step Solution

Answer

\( \log\left(\frac{x^{1/2}}{y^{1/3}z^2}\right) \)

Step 1: Apply Power Rule of Logarithms

The power rule of logarithms states that \( k \, \log a = \log(a^k) \). Apply this to each term:- \( \frac{1}{2} \log x = \log(x^{1/2}) \)- \( \frac{1}{3} \log y = \log(y^{1/3}) \)- \( 2 \log z = \log(z^2) \).

Step 2: Substitute Transformed Logarithm Terms

Replace each term in the original expression with its transformed version from Step 1:\[ \log(x^{1/2}) - \log(y^{1/3}) - \log(z^2) \]

Step 3: Apply the Quotient Rule of Logarithms

The quotient rule states that \( \log a - \log b = \log\left(\frac{a}{b}\right) \). Apply this rule pairwise:First, combine the first two terms:\[ \log(x^{1/2}) - \log(y^{1/3}) = \log\left(\frac{x^{1/2}}{y^{1/3}}\right) \]Next, include the third term:\[ \log\left(\frac{x^{1/2}}{y^{1/3}}\right) - \log(z^2) = \log\left(\frac{x^{1/2}}{y^{1/3} \cdot z^2}\right) \]

Step 4: Rewrite as a Single Logarithm

This final expression is already a single logarithm:\[ \log\left(\frac{x^{1/2}}{y^{1/3} \cdot z^2}\right) \]This is now expressed as a single logarithm with a coefficient of 1.

Key Concepts

Properties of Logarithms

Logarithms have several key properties that make them useful for simplifying and manipulating mathematical expressions. Understanding these properties will help you grasp how logarithms work.
1. **Product Rule:** This rule states that the logarithm of a product is the sum of the logarithms. Mathematically, this can be expressed as: \( \log(ab) = \log(a) + \log(b) \).
2. **Quotient Rule:** According to the quotient rule, the logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).
3. **Power Rule:** This property states that the logarithm of a power is the exponent times the logarithm of the base. For example: \( \log(a^b) = b \log(a) \).
These properties are indispensable when rewriting and solving logarithmic expressions. They allow us to simplify complex equations into more manageable forms.

Power Rule of Logarithms

The power rule of logarithms can seem a bit tricky at first. However, it becomes intuitive once you understand its utility. This rule states that you can bring down an exponent as a coefficient in front of the logarithm. Formally, it's written as \( b \log(a) = \log(a^b) \).
In our exercise, we used this rule to transform each term in the expression:

• \(\frac{1}{2} \log(x)\) was rewritten as \(\log(x^{1/2})\), essentially as \(\log(\sqrt{x})\).
• \(\frac{1}{3} \log(y)\) became \(\log(y^{1/3})\) or \(\log(\sqrt[3]{y})\).
• \(2 \log(z)\) was transformed into \(\log(z^2)\).

These simplifications make it easier to work with the full expression using further properties or mathematical operations.

Quotient Rule of Logarithms

The quotient rule of logarithms is another useful tool when simplifying equations. It allows you to combine or break apart logarithms in relation to division. The rule is expressed as: \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \).
In practice, this means when you subtract the logarithm of one number from the logarithm of another, it's equivalent to taking the logarithm of their quotient.
In our exercise:

• First, the expression \(\log(x^{1/2}) - \log(y^{1/3})\) was simplified into \(\log\left(\frac{x^{1/2}}{y^{1/3}}\right)\).
• Then, combining this with the third term results in \(\log\left(\frac{x^{1/2}}{y^{1/3} \cdot z^2}\right)\), further simplifying it to a more compact, single logarithm form.

Understanding these rules allows you to work efficiently with logarithmic expressions, leading to elegant solutions.