Problem 3 - Precalculus: Functions and Graphs

Question

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log _{x} \frac{x^{3} w}{y^{2} z^{4}}$$

Step-by-Step Solution

Answer

\(3 + \log_{x}(w) - 2 \log_{x}(y) - 4 \log_{x}(z)\)

Step 1: Apply the Quotient Rule

The logarithm of a quotient can be expressed as the difference of the logarithms of the numerator and the denominator. Apply this rule to get:\[\log_{x} \left(\frac{x^{3} w}{y^{2} z^{4}}\right) = \log_{x}(x^3 w) - \log_{x}(y^2 z^4)\]

Step 2: Apply the Product Rule

Use the product rule for logarithms, which states that the logarithm of a product is the sum of the logarithms, to both the numerator and the denominator:\[\log_{x}(x^3 w) = \log_{x}(x^3) + \log_{x}(w)\]\[\log_{x}(y^2 z^4) = \log_{x}(y^2) + \log_{x}(z^4)\]

Step 3: Simplify Using the Power Rule

Apply the power rule for logarithms, which says that \( \log_{a}(b^c) = c \log_{a}(b) \), to simplify each term:\[\log_{x}(x^3) = 3 \log_{x}(x)\]\[\log_{x}(y^2) = 2 \log_{x}(y)\]\[\log_{x}(z^4) = 4 \log_{x}(z)\]

Step 4: Simplify the Expression

Since \( \log_{x}(x) = 1 \), replace this in the expression and combine all the terms:\[3 \log_{x}(x) + \log_{x}(w) - (2 \log_{x}(y) + 4 \log_{x}(z))\]Simplifies to:\[3 + \log_{x}(w) - 2 \log_{x}(y) - 4 \log_{x}(z)\]

Key Concepts

Quotient Rule for Logarithms

When dealing with logarithms, the quotient rule is incredibly useful to simplify complex expressions. The quotient rule states that the logarithm of a fraction can be broken down into a subtraction:

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

This means if you have a division inside a logarithm, you can split it into two separate logarithm terms. Each of these terms reflects the numerator and the denominator, respectively.
In our specific exercise, the logarithm \( \log_{x} \left( \frac{x^{3} w}{y^{2} z^{4}} \right) \) involved applying the quotient rule. By translating this into \( \log_{x}(x^3 w) - \log_{x}(y^2 z^4) \), you make it easier to further simplify each part individually. This is the foundational step that allows for the expression to be tackled with additional logarithmic rules.

Product Rule for Logarithms

The product rule for logarithms helps further simplify expressions. This rule states that the logarithm of a product is equal to the sum of the logarithms:

• \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \)

This rule is handy when you have multiplication inside a logarithm, as it allows you to split it into several easier-to-manage parts.
In our exercise, both the numerator \( x^3 w \) and the denominator \( y^2 z^4 \) are products. By applying the product rule:

• \( \log_{x}(x^3 w) = \log_{x}(x^3) + \log_{x}(w) \)
• \( \log_{x}(y^2 z^4) = \log_{x}(y^2) + \log_{x}(z^4) \)

This breaks down the complex expression into simpler terms, allowing for easier application of other rules, like the power rule, to further simplify each component.

Power Rule for Logarithms

The power rule for logarithms is especially useful for simplifying expressions where exponents are involved. It states that you can bring the exponent in front of the logarithm as a multiplier:

• \( \log_{b}(m^n) = n \cdot \log_{b}(m) \)

This rule transforms a potentially complex power expression to a simpler arithmetic product.
In the solution for our exercise, the power rule was crucial. Each part of the expression that contained a power was simplified using this rule:

• \( \log_{x}(x^3) = 3 \cdot \log_{x}(x) \)
• \( \log_{x}(y^2) = 2 \cdot \log_{x}(y) \)
• \( \log_{x}(z^4) = 4 \cdot \log_{x}(z) \)

This effective simplification step allows for the equation to be combined into a manageable expression. Furthermore, knowing that \( \log_{x}(x) = 1 \), you can substitute and sum up all terms, finalizing the simplification into the neat expression: \( 3 + \log_{x}(w) - 2 \log_{x}(y) - 4 \log_{x}(z) \).