Problem 78 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(-2 \log x-3 \log y+\log z\)

Step-by-Step Solution

Answer

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

Step 1: Apply the Power Rule

Use the power rule of logarithms, which states that \(^a \log b = \log b^a\), to move the coefficients as exponents on their respective arguments. Thus, the expression becomes: \(\log x^{-2} + \log y^{-3} + \log z\).

Step 2: Combine Using the Product Rule

Use the product rule of logarithms, which states that \(\log a + \log b = \log (a \cdot b)\). Combine the three terms into one logarithm: \(\log (x^{-2} \cdot y^{-3} \cdot z)\).

Step 3: Simplify the Expression

The expression inside the logarithm can be further simplified: \(x^{-2} \cdot y^{-3} \cdot z = \frac{z}{x^2 \cdot y^3}\). Therefore, the final expression is: \(\log \left( \frac{z}{x^2 \cdot y^3} \right)\).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a useful tool that allows us to manipulate expressions where a logarithm is multiplied by a constant. It helps convert the multiplication into an exponent, making the expression easier to work with. For example, if you have an expression like \(-2 \log x\), applying the power rule makes it \(\log x^{-2}\). This rule is formally written as: \( a \log b = \log b^a\).
Here's how it works:

• The constant (like the \(-2\) in \(-2 \log x\)) becomes the exponent of the argument of the logarithm.
• This helps simplify further operations like addition or subtraction of logarithmic terms.

By knowing how to utilize this rule, you'll find it easier to transition to other rules and steps in simplifying expressions.

Product Rule of Logarithms

The product rule of logarithms helps you combine multiple logarithmic terms into a single expression. It states that the sum of logs is the log of the product. It is formally written as: \( \log a + \log b = \log (a \cdot b) \).
In practical terms, it means:

• When you have two or more logarithms with the same base being added, you can combine them into a single logarithm representing the product of their arguments.
• This rule simplifies the expression and makes it easier to manage and further simplify.

For instance, given \(\log x^{-2} + \log y^{-3} + \log z\), you can apply the product rule to combine them into one logarithm: \(\log (x^{-2} \cdot y^{-3} \cdot z)\) . This is a crucial step in consolidating and simplifying logarithmic expressions.

Simplifying Logarithms

Once you have used the rules of logarithms to combine and manipulate the expression, the next step is simplifying the terms inside the logarithm. Simplification often involves reducing the expression to a simpler, more manageable form. For example, after applying the rules, if you have \(\log (x^{-2} \cdot y^{-3} \cdot z)\), you can simplify what's inside the logarithm:

• You rewrite this as a fraction: \(\frac{z}{x^2 \cdot y^3}\). This is done by recognizing the negative exponents.
• By converting multiplication into division using negative exponents, it becomes easier to see the simplest form of the expression.

This step is about expressing the product of variables in the simplest form, making it easier to understand and work with. Understanding how to effectively simplify logarithmic expressions will enhance your problem-solving skills and clarity in dealing with exponential and logarithmic problems.