Problem 21 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x^{2} y\right) $$

Step-by-Step Solution

Answer

The logarithmic expression \(\log_{b}(x^{2} y)\) expands to \(2 * \log_{b}(x) + \log_{b}(y)\).

Step 1: Identify The Seasoned Logarithmic Expression

Here, we have a logarithmic expression \(\log_{b}(x^{2} y)\), which is a multi-parameter logarithm equation. This can be broken down into individual parts due to the properties of logarithm.

Step 2: Apply the Property of Logarithms

The logarithm of a product, \(\log_{b}(\(xy\))\), is equal to the sum of the logarithms of \(x\) and \(y\). Applying this property to our expression, we get \(\log_{b}(x^{2}) + \(\log_{b}(y)\).

Step 3: Further Expansion

Moreover, another property is that the logarithm of an exponent, \(\log_{b}(x^{n})\), is equal to the product of \(n\) and the logarithmic function of the given variable \(\log_{b}(x)\). So we further expand \(\log_{b}(x^{2})\) to \(2 * \log_{b}(x)\). Thus our full expanded expression becomes \(2 * \log_{b}(x) + \log_{b}(y)\).

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is crucial for solving various mathematical problems. A logarithm can be thought of as the opposite of an exponent. If you have an exponential equation like \(b^x = y\), the logarithmic form would be \(\log_b(y) = x\), where \(b\) is the base, \(y\) is the result, and \(x\) is the exponent to which the base must be raised to produce \(y\). Logarithmic expressions involve the operation of taking logarithms and usually incorporate variables and constants. Simplifying logarithmic expressions is essential, and this involves using properties of logarithms to express complex expressions in terms of simpler ones.

The main challenge for students is to recognize the patterns that allow the use of logarithmic properties. For example, when given an expression like \(\log_b(x^2y)\), it's important to notice that it consists of a product within the logarithm, allowing for expansion using specific logarithmic properties.

Expanding Logarithms

Expanding logarithms means to rewrite a logarithmic expression that usually involves compound arguments (like products or powers) into a series of simpler, additive logarithmic terms. The ability to expand logarithms is useful when solving equations or simplifying complex expressions. Expanding logarithms makes use of key properties like the product, quotient, and power rules.

One common struggle is identifying when and how to apply these rules correctly. To do this effectively, first look for exponents, products, or quotients within the logarithm. Our example \(\log_b(x^2y)\) contains both a power and a product, indicating that we can expand the expression using the relevant properties. Ensuring each term is fully expanded involves repeated application of these properties until no further expansion is possible.

Logarithm of a Product

The logarithm of a product property is intuitive once you understand the fundamental nature of logarithms. It states that the logarithm of a product of two numbers is the sum of the logarithms of the individual numbers. Mathematically, it's expressed as \(\log_b(xy) = \log_b(x) + \log_b(y)\).

This property is especially helpful when you encounter a logarithm that encompasses a multiplication within its argument, as it simplifies complex expressions into additive terms, which are much easier to handle. In practice, when you come across an expression like \(\log_b(x^2y)\), this property tells you that the logarithm of the product \(x^2y\) can be written as the sum of the logarithms of \(x^2\) and \(y\), simplifying the expression.

Logarithm of an Exponent

Logarithms have a unique relationship with exponents, which is highlighted in the logarithm of an exponent rule. This rule states that the logarithm of a number raised to an exponent is the same as the exponent times the logarithm of the base number or \(\log_b(x^n) = n \log_b(x)\).

When you see an expression like \(\log_b(x^2)\), this property allows you to bring the exponent out front, multiplying it by the logarithm of the base. It turns the expression into something much simpler: \(2 \log_b(x)\). Applying this property streamlines the process of simplifying logarithmic expressions and is a key tool for fully expanding logarithms. Remember, it is the exponent on the argument of the logarithm that gets moved, not the base of the logarithm.