Problem 1 - Precalculus: Functions and Graphs

Question

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\text { (a) } \log _{4}(x z)$$ (b) \(\log _{4}(y / x)\) (c) \(\log _{4} \sqrt[3]{z}\)

Step-by-Step Solution

Answer

(a) \(\log_4(x) + \log_4(z)\), (b) \(\log_4(y) - \log_4(x)\), (c) \(\frac{1}{3} \log_4(z)\).

Step 1: Review Logarithm Product Rule

The logarithm product rule states that \( \log_b(mn) = \log_b(m) + \log_b(n) \). This rule will be useful for part (a) to separate the variables inside the logarithm.

Step 2: Apply Product Rule to (a)

For (a), \( \log_{4}(xz) \), we can apply the product rule: \( \log_{4}(x) + \log_{4}(z) \). This expresses the logarithm in terms of \(x\) and \(z\).

Step 3: Review Logarithm Quotient Rule

The logarithm quotient rule states that \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). This rule will help simplify part (b).

Step 4: Apply Quotient Rule to (b)

For (b), \( \log_{4}(\frac{y}{x}) \), use the quotient rule: \( \log_{4}(y) - \log_{4}(x) \). This provides an expression in terms of \(y\) and \(x\).

Step 5: Review Logarithm Power Rule

The logarithm power rule is \( \log_b(m^n) = n \cdot \log_b(m) \). We can use this rule for part (c) to handle the cube root.

Step 6: Apply Power Rule to (c)

For (c), \( \log_{4}(\sqrt[3]{z}) \), recognize it as \(z^{1/3}\). Use the power rule to get \( \frac{1}{3} \cdot \log_{4}(z) \).

Key Concepts

Logarithm Product Rule

When dealing with products inside a logarithm, the Product Rule for logarithms is your friend. This rule tells us that the logarithm of a product is simply the sum of the logarithms of the factors. For instance, if we have

• \( \log_b(mn) \),

it can be broken down to

• \( \log_b(m) + \log_b(n) \).

Why does this work? It springs from the properties of exponents, where multiplying same-base exponents leads to adding the exponents. Similarly, a logarithm is an exponent. So, if you're faced with something like \( \log_4(xz) \), you can separate it into the sum of \( \log_4(x) \) and \( \log_4(z) \). Use this technique to simplify logarithmic expressions with products. Remember that it applies to any number of factors contained within a single logarithm.

Logarithm Quotient Rule

Logarithms offer a straightforward way to handle division through the Quotient Rule. The essence of this rule is that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In a mathematical form, this can be expressed as:

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

This comes from the logical parallel with the division of powers, where dividing corresponds to subtracting exponents. For example, if you encounter an expression like \( \log_4\left(\frac{y}{x}\right) \), you simplify it to \( \log_4(y) - \log_4(x) \). This rule is a powerful tool in breaking down complex fractional expressions within a logarithm, making them easier to work with.

Logarithm Power Rule

The Power Rule for logarithms sheds light on how to deal with exponents within a logarithmic expression. This useful property states that the logarithm of a power is the exponent times the logarithm of the base. It is mathematically defined as:

• \( \log_b(m^n) = n \cdot \log_b(m) \).

The rationale here echoes the principle of raising powers, which multiplies exponents. Applying this rule, take a scenario such as \( \log_4(\sqrt[3]{z}) \). Recognizing \( \sqrt[3]{z} \) as \( z^{1/3} \), you can simplify the expression to \( \frac{1}{3} \cdot \log_4(z) \). This rule is particularly handy when you need to simplify expressions involving roots or powers within logarithms.