Problem 271 - Precalculus

Question

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ \log (x)-\frac{1}{2} \log (y)+3 \log (z) $$

Step-by-Step Solution

Answer

\( \log\left(\frac{xz^3}{\sqrt{y}}\right) \)

Step 1: Apply logarithm power property

The logarithm power property states that \( a \log(b) = \log(b^a) \). Using this, we can rewrite each term in the expression: \( \frac{1}{2} \log(y) \) becomes \( \log(y^{1/2}) \) and \( 3 \log(z) \) becomes \( \log(z^3) \).

Step 2: Rewrite the expression with power property

Substitute the transformed terms back into the expression: \( \log(x) - \log(y^{1/2}) + \log(z^3) \).

Step 3: Combine logs using product property

The logarithm product property states \( \log(a) + \log(b) = \log(ab) \). Using this property, combine \( \log(x) + \log(z^3) \) to get \( \log(xz^3) \).

Step 4: Combine logs using quotient property

The logarithm quotient property states \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \). Apply this property to \( \log(xz^3) - \log(y^{1/2}) \), resulting in \( \log\left(\frac{xz^3}{y^{1/2}}\right) \).

Key Concepts

Understanding Logarithm Power Property

The logarithm power property is a fundamental concept that simplifies expressions involving logarithms. It allows us to move a coefficient in front of a log as an exponent, creating an expression that's easier to manage.
For example, if you have an expression like \( a \log(b) \), you can rewrite it as \( \log(b^a) \).
This transformation uses the power property to simplify the process of further combining logs.
Consider a few common scenarios:

• \( 2 \log(x) \) becomes \( \log(x^2) \)
• \( \frac{1}{2} \log(y) \) can be transformed into \( \log(y^{1/2}) \)
• \( -3 \log(z) \) would become \( \log(z^{-3}) \)

This property is incredibly helpful when you're trying to condense or manipulate logarithmic expressions for mathematical operations or solving equations.

Decoding Logarithm Product Property

Another vital tool is the logarithm product property. It provides a means to merge two logarithms into a single expression by linking them through multiplication.
When you have a situation like \( \log(a) + \log(b) \), the product property lets you combine these into \( \log(ab) \).
This is particularly useful when consolidating terms for easier calculation or comparison.
Visualizing this through examples:

• \( \log(3) + \log(5) = \log(3 \times 5) = \log(15) \)
• \( \log(m) + \log(n) = \log(mn) \)
• \( \log(x^2) + \log(y) = \log(x^2y) \)

With this property, your task of simplifying expressions becomes more straightforward, as you can immediately see the connection through multiplication!

Exploring Logarithm Quotient Property

The logarithm quotient property acts as the counterpart to the product property, allowing for the simplification of expressions involving division.
Understanding this property helps when you have an expression like \( \log(a) - \log(b) \) because you can condense it into \( \log\left(\frac{a}{b}\right) \).
This conversion is crucial when working towards a simpler logarithmic form or solving equations.
Here are some illustrative cases:

• \( \log(15) - \log(3) = \log\left(\frac{15}{3}\right) = \log(5) \)
• \( \log(x) - \log(y) = \log\left(\frac{x}{y}\right) \)
• \( \log(m^2) - \log(n^3) = \log\left(\frac{m^2}{n^3}\right) \)

By applying the logarithm quotient property, you can streamline complex expressions, making them more manageable and easier to work with.