Problem 79 - Algebra 2

Question

Expand each logarithm. \(\log \left(\frac{2 \sqrt{x}}{5}\right)^{3}\)

Step-by-Step Solution

Answer

The expanded form of \(\log \left(\frac{2 \sqrt{x}}{5}\right)^{3}\) is \(3 \log 2 + 3 \log \sqrt{x} - 3 \log 5\).

Step 1: Use the Power Rule

The Power Rule of logarithms says that any exponent in the argument of a logarithm can be moved out front as a multiplier. Therefore, \(\log \left(\frac{2 \sqrt{x}}{5}\right)^{3}\) becomes \(3 \log \left(\frac{2 \sqrt{x}}{5}\right)\).

Step 2: Apply the Quotient Rule

The Quotient Rule of logarithms states that the logarithm of a quotient can be written as the difference of two logs. So, \(3 \log \left(\frac{2 \sqrt{x}}{5}\right)\) becomes \(3 (\log(2 \sqrt{x}) - \log(5))\).

Step 3: Utilize the Product Rule

The Product Rule says that the log of a product equals the sum of the logs. So within the first log, separate \(2 \sqrt{x}\) into \(\log 2 + \log \sqrt{x}\). Therefore, the fully expanded log is \(3 (\log 2 + \log \sqrt{x} - \log 5)\).

Step 4: Distribute the Multiplier

Now distribute the 3 into the brackets, resulting in \(3 \log 2 + 3 \log \sqrt{x} - 3 \log 5\).

Key Concepts

Power Rule of Logarithms

The Power Rule of logarithms is a fundamental principle that greatly simplifies the handling of logarithms involving exponents. This rule states that an exponent on an argument inside a logarithmic function can be moved to the front as a coefficient. For instance, if you have a logarithm of the form \( \log_b (a^n) \), you can rewrite it as \( n \cdot \log_b (a) \).

This rule works because of the properties of exponents and logarithms. Since logarithms are the inverse of exponents, moving the exponent out allows us to handle it more conveniently. For example, in the expression \( \log \left(\frac{2 \sqrt{x}}{5}\right)^{3} \), you effectively turn the problem of raising a complex expression to a power into a simpler multiplication by a constant factor—3 in this case.

By applying the Power Rule, the complicated-looking expression is reduced drastically to \( 3 \log \left(\frac{2 \sqrt{x}}{5}\right) \), setting up the next steps for further breakdown using other logarithmic rules.

Quotient Rule of Logarithms

The Quotient Rule of logarithms is a helpful tool for breaking down a logarithm that includes a fraction. This rule states that the logarithm of a quotient can be expressed as the difference of the logarithms of the numerator and the denominator. Mathematically, this is represented as \( \log_b \left(\frac{a}{c}\right) = \log_b (a) - \log_b (c) \).

When you encounter a fraction inside a logarithmic function, this rule helps by allowing you to separate it into more manageable parts. Take, for example, the expression derived from our exercise: \( 3 \log \left(\frac{2 \sqrt{x}}{5}\right) \). According to the Quotient Rule, this can be factored into \( 3 ( \log(2 \sqrt{x}) - \log(5) ) \).

This separation is valuable for simplifying logarithmic expressions, making complex calculations simpler. It fundamentally changes a single logarithm with a division to two separate logarithms with a subtraction, preparing the ground for further simplification using the Product Rule.

Product Rule of Logarithms

The Product Rule of logarithms is another handy property that lets you break down the log of a product into a sum of logs. Specifically, if you have an expression \( \log_b (a \cdot c) \), it can be rewritten as \( \log_b (a) + \log_b (c) \).

This rule is particularly useful when dealing with expressions where multiple components are multiplied together, as it allows you to consider each part separately. In our example, after applying the Quotient Rule, we have \( 3 (\log(2 \sqrt{x}) - \log(5)) \). Within the first term, \( \log(2 \sqrt{x}) \), we can further expand it using the Product Rule: \( \log(2 \cdot \sqrt{x}) = \log(2) + \log(\sqrt{x}) \).

The result is that the complex product now becomes a straightforward sum of other logarithms, namely \( 3 (\log 2 + \log \sqrt{x} - \log 5) \). This breakdown into simpler parts greatly aids in both computation and understanding, making complex logarithmic expressions more accessible.