Problem 4 - Algebra and Trigonometry

Question

(a) We can expand \(\log \left(\frac{x^{2} y}{z}\right)\) to get _____. (b) We can combine \(2 \log x+\log y-\log z\) to get _____.

Step-by-Step Solution

Answer

(a) The expansion is: \(2 \log x + \log y - \log z\). (b) Combination gives \(\log\left( \frac{x^2 y}{z} \right)\).

Step 1: Use Logarithm Properties for Part (a)

To expand the expression \( \log \left( \frac{x^2 y}{z} \right) \), we utilize the properties of logarithms. Begin by using the quotient rule: \( \log \left( \frac{A}{B} \right) = \log A - \log B \). This gives us \( \log(x^2 y) - \log z \).

Step 2: Further Expand Using Logarithm Properties for Part (a)

Next, apply the product rule to the numerator \( \log(x^2 y) \): \( \log(x^2 y) = \log x^2 + \log y \). Substitute back into the expression from Step 1: \( \log x^2 + \log y - \log z \).

Step 3: Simplify Using the Power Rule for Part (a)

Apply the power rule \( \log(x^a) = a \log x \) to \( \log x^2 \). Thus, the expression becomes \( 2 \log x + \log y - \log z \). Hence, the expanded form is \( 2 \log x + \log y - \log z \).

Step 4: Verify the Combined Expression for Part (b)

The expression provided in part (b) is \( 2 \log x + \log y - \log z \), which matches what we found in part (a). Using the rules of logarithms in reverse, combine these terms back into a single logarithm: \( \log\left( x^2 y \right) - \log z \).

Step 5: Use Logarithm Properties for Part (b)

Following the properties, combine the terms: \( \log\left( \frac{x^2 y}{z} \right) \). This verifies that combining the expression gives \( \log\left( \frac{x^2 y}{z} \right) \).

Key Concepts

Logarithm Expansion

Logarithm expansion is a method to break down complex logarithmic expressions into simpler parts. This process utilizes various logarithm properties to express a single logarithm as a combination of simpler logarithmic terms. For example, given an expression like \( \log \left( \frac{x^2 y}{z} \right) \), you can expand it by applying different logarithmic rules.

The first step is to consider the denominator and the numerator separately using the quotient rule. This tells us that \( \log \left( \frac{A}{B} \right) = \log A - \log B \). Now, let's take a closer look at the numerator, replacing its products with addition by using the product rule: \( \log(x^2 y) = \log x^2 + \log y \). Each piece of this expanded expression can then be further simplified or manipulated using other properties.

Power Rule

The power rule is a handy logarithmic property used when you have a logarithm with an exponent. This rule states that \( \log_b(x^a) = a \log_b(x) \), allowing you to bring the exponent down in front of the logarithm as a multiplier.

In the context of our previous problem, where you have \( \log(x^2) \), applying the power rule transforms this into \( 2 \log x \). This simplifies the expression and is crucial when expanding logarithms because it reduces the complexity of the power term.

• Helps in simplifying logarithmic expressions
• Transforms power terms into linear components

By consistently applying the power rule, you can easily handle coefficients and combine different logarithms together in a more manageable form.

Quotient Rule

The quotient rule is a core component when working with logarithmic expansions and simplifications. It is expressed as \( \log \left( \frac{A}{B} \right) = \log A - \log B \). This rule aids in separating a complicated fraction inside a logarithm into subtractive form.

For example, in the expression \( \log \left( \frac{x^2 y}{z} \right) \), the quotient rule quickly shows us how to separate the terms: first breaking it into \( \log(x^2 y) - \log z \). This is the foundational step in expanding or simplifying logarithmic expressions involving fractions. Without the quotient rule, log problems with fractions would remain unnecessarily cumbersome.

Product Rule

The product rule simplifies logarithmic expressions involving products within the logarithm. Essentially, this rule states that \( \log(AB) = \log A + \log B \). It lets you convert products into sums, making it easier to handle the expressions using further properties.

Taking the expression \( \log(x^2 y) \) from our example, the product rule allows us to split it into two separate terms: \( \log x^2 + \log y \). Post-application, each component term can be independently analyzed or simplified further, such as by applying the power rule to \( \log x^2 \).

• Converts multiplicative log terms into addition
• Facilitates easier simplification and manipulation of expressions

Using the product rule in tandem with other logarithm properties creates a straightforward path to solving complex logarithmic equations.