Problem 56 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{10} \frac{x y^{4}}{z^{5}}$$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(\log_{10}(x) + 4 \log_{10}(y) - 5 \log_{10}(z)\)

Step 1: Apply Logarithmic Property of Quotient

The initial expression can be rewritten as a difference of two logs applying logarithm property of quotient. So, \(\log_{10} \frac{x y^{4}}{z^{5}} = \log_{10}(x y^{4}) - \log_{10}(z^{5})\)

Step 2: Apply Logarithmic Property of Product

In the first part of expression, we have \(x y^{4}\) (a multiplication). The logarithm property of product allows to split it into a sum of two separate logs. Therefore, \(\log_{10}(x y^{4}) = \log_{10}(x) + \log_{10}(y^{4})\). Replace this into the expression obtained in the previous step

Step 3: Apply Logarithmic Property of Exponents

To finalize the expression, apply the logarithm property of exponent, which allows to move the exponent in front: \(\log_{10}(y^{4}) = 4 \log_{10}(y)\) and \(\log_{10}(z^{5}) = 5 \log_{10}(z)\). Replace these into expression obtained in previous steps, to get the final expression

Key Concepts

Properties of Logarithms

Understanding the various properties of logarithms is essential when dealing with logarithmic expressions. Essentially, logarithms are exponents, and these properties help simplify complex expressions into more manageable forms. Here are some key principles to keep in mind:

• The logarithm of a product can be written as the sum of the logarithms of the individual factors.
• The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.
• The logarithm of a power can be expressed as the exponent times the logarithm of the base.

By deftly applying these rules, mathematicians and students alike can expand, condense, or solve logarithmic equations step by step, making them more suitable for further calculations or making them easier to understand.

Logarithmic Property of Quotient

Let's take a closer look at the logarithmic property of quotient, which is pivotal when expanding logarithmic expressions. Generally stated, the logarithm of a quotient is the difference of the logarithms of the numerator and denominator:
\begin{align*}\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\end{align*}Where \(b\) is the base of the logarithm, \(M\) is the numerator, and \(N\) is the denominator. This property is directly used to break down complex divisions within a log into a subtraction of two simpler logs, making them more accessible for both understanding and computation.

Logarithmic Property of Product

Expanding on the properties, let's delve into the logarithmic property of product. This property states that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers:\begin{align*}\log_b(M \cdot N) = \log_b(M) + \log_b(N)\end{align*}In practice, if you encounter a multiplication inside a logarithmic function, you can separate it into the addition of two logs. This can greatly simplify the original problem. For example, if you were to apply this property to \(\log_b(xy)\), it would become \(\log_b(x) + \log_b(y)\). This step is quite common when expanding logarithms and makes solving or simplifying them much less intimidating.

Logarithmic Property of Exponents

Finally, we have the logarithmic property of exponents, a cornerstone for manipulating logarithmic expressions involving powers. The rule is straightforward:
\begin{align*}\log_b(M^p) = p \cdot \log_b(M)\end{align*}This means you can 'pull out' the exponent from inside the log and multiply it in front. This is particularly useful when dealing with expressions like \(\log_b(y^4)\), transforming it simply into \(4 \log_b(y)\). This property demonstrates the deep connection between exponents and logarithms and is a tool that simplifies otherwise complex logarithmic expressions into something far more digestible.