Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

Express each part using these logarithm properties.

Step 1: Part (a)

\(\log_4(xz) = \log_4 x + \log_4 z\) (product rule of logarithms).

Step 2: Part (b)

\(\log_4\left(\frac{y}{x}\right) = \log_4 y - \log_4 x\) (quotient rule of logarithms).

Step 3: Part (c)

\(\log_4 \sqrt[3]{z} = \log_4 z^{1/3} = \frac{1}{3}\log_4 z\) (power rule of logarithms).

Key Concepts

Properties of Logarithms

Logarithms are mathematical tools that help us simplify complex expressions, especially when dealing with multiplications, divisions, and powers. In this section, we'll talk about the foundational properties of logarithms:
1. **Product Property**: When you take the logarithm of a product, say two numbers or expressions multiplied together, it transforms into the sum of two separate logarithms. Mathematically, that's written as \( \log_b(mn) = \log_b(m) + \log_b(n) \). This property simplifies problems involving products, making them easier to manage with basic addition.
2. **Quotient Property**: This property is quite handy when you're dealing with division inside a log. It states that the logarithm of a quotient can be broken down into the difference of two logarithms: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). Using this, you can switch from division to subtraction, which is much simpler to handle.
3. **Power Property**: When you have a power within a logarithm, you can bring the exponent out in front, turning it into a multiplication: \( \log_b(m^n) = n\log_b(m) \). This way, exponents become coefficients, and you switch from dealing with powers to straightforward multiplication.
These properties make working with logs much more straightforward, allowing complex calculations to be approached methodically.

Logarithmic Expressions

Logarithmic expressions are composed of logarithms, their arguments, and sometimes coefficients or exponents. Understanding these expressions is key to manipulating and rewriting them, often using the properties of logarithms.

• When faced with a logarithmic expression like \( \log_b(xz) \), you use the product property to express it as \( \log_b(x) + \log_b(z) \). This breaks down a multiplication inside the log into simpler parts.
• Similarly, \( \log_b(y/x) \) can be rewritten using the quotient property as \( \log_b(y) - \log_b(x) \), making it easier to work with by turning division into subtraction.
• If you have something like \( \log_b(\sqrt[3]{z}) \), you can use the power property. Recognize that the cube root of \( z \) is \( z^{1/3} \) and rewrite the log as \( \frac{1}{3}\log_b(z) \).

By breaking down these expressions using the properties of logarithms, you can transform them into simpler, more manageable forms. This is extremely helpful for simplifying complex equations or solving logarithmic equations.

Logarithmic Identities

Logarithmic identities are useful relationships between logarithms that hold universally, regardless of the specific values involved. These identities stem from the properties of logarithms and can be applied to simplify expressions or solve equations.
Some essential logarithmic identities include:

• **Change of Base Formula**: This states that any logarithm can be rewritten as a fraction of two logarithms of any base. If you have \( \log_b(a) \), it can also be expressed as \( \frac{\log_c(a)}{\log_c(b)} \). This is especially useful for calculating logs with bases not supported by calculators.
• **Zero Logarithm Identity**: In any logarithmic system, \( \log_b(1) = 0 \). Since any number to the power of zero is one, the logarithm of any base to one is zero.
• **Identity of Logarithm**: Another simple yet powerful identity is, \( \log_b(b) = 1 \). This tells us that the log of a base to itself equals one, as it is the power you raise the base to get itself.

These identities are like shortcuts that make working with logarithms quicker and less cumbersome. They can save time and reduce potential mistakes when solving complex logarithmic problems.