Problem 34 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right)\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \( \frac{1}{3}\log_b(x) + 4\log_b(y) - 5\log_b(z) \).

Step 1: Apply the division property of logarithms

Start by applying the division property of logarithms which says that \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). In this case the expression can be split into: \( \log_b(\sqrt[3]{x}y^4) - \log_b(z^5) \).

Step 2: Apply the multiplication property of logarithms

Next, apply the multiplication property of logarithms which says that \( \log_b(mn) = \log_b(m) + \log_b(n) \). This is applied to the first logarithmic term from Step 1 which results in: \( \log_b(\sqrt[3]{x}) + \log_b(y^4) - \log_b(z^5) \).

Step 3: Apply the power property of logarithms

Now the power properties of logarithms can be utilised, which say that \( \log_b(m^n) = n\log_b(m) \). This is applied to each of the logarithmic expression obtained in Step 2, which gives: \( \frac{1}{3}\log_b(x) + 4\log_b(y) - 5\log_b(z) \).

Key Concepts

Division Property of Logarithms

Logarithms are a fascinating mathematical tool often used to simplify complex expressions. One vital property is the division property of logarithms. This concept is essential for expanding logarithmic expressions to make them more manageable. The division property states that for any positive numbers \(m\) and \(n\), and a base \(b\):

• \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)

This means when you have a quotient inside a logarithm, you can separate it into the difference of two logs.
Think of it like turning a complicated division inside the log into two simpler subtractions!
For example, if you have the expression \(\log_b\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right)\), you would start by splitting it using this property:

• \(\log_b(\sqrt[3]{x} y^{4}) - \log_b(z^{5})\)

This is the essential first step in breaking down the problem, making it easier to work with. Perfect for reducing complexity in your calculations!

Multiplication Property of Logarithms

After using the division property, the next logical step is to handle any multiplication inside the logarithm. This is where the multiplication property of logarithms comes into play. It helps us further expand the terms that are multiplied together under a log.
The basic formula for the multiplication property is:

• \(\log_b(mn) = \log_b(m) + \log_b(n)\)

This property allows you to break up the product inside the logarithm into the sum of two separate logs.
It's like unraveling a bundle of ropes into individual strands.
For the expression \(\log_b(\sqrt[3]{x} y^{4})\), apply this property to get:

• \(\log_b(\sqrt[3]{x}) + \log_b(y^4)\)

This step-by-step application ensures each component is clearly isolated, helping you spot further simplifications or properties you can employ in later steps.

Power Property of Logarithms

Finally, once you've dealt with division and multiplication, you can use the power property of logarithms to fully expand your expression. This property is crucial because it changes terms raised to a power into something much simpler and easier to handle. The power property states:

• \(\log_b(m^n) = n \log_b(m)\)

This effectively transforms the exponent into a multiplier outside the log.
Imagine it as extracting the power and placing it neatly in front!
Apply this property to the terms in the expression you've been working on:

• \(\log_b(\sqrt[3]{x}) = \frac{1}{3}\log_b(x)\)
• \(\log_b(y^4) = 4\log_b(y)\)
• \(\log_b(z^5) = 5\log_b(z)\)

Thus, your final expanded expression is:

• \(\frac{1}{3}\log_b(x) + 4\log_b(y) - 5\log_b(z)\)

Applying this property offers clarity and conciseness. It ensures mathematical expressions are in a format that's easy to interpret, especially when preparing for further algebraic manipulations or calculations.