Problem 23 - 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. $$ \log _{4}\left(\frac{\sqrt{x}}{64}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(0.5 \times \log_{4}{x} - 3\).

Step 1: Apply the Quotient Rule

Firstly, apply the quotient rule which states that the logarithm of a division is equal to the difference of the logarithm of the numerator and the logarithm of the denominator. Thus, \( \log _{4}\left(\frac{\sqrt{x}}{64}\right) \) can be rewritten as \( \log_{4}{\sqrt{x}} - \log_{4}{64} \).

Step 2: Apply the Power Rule

Next, apply the power rule which states that the logarithm of an expression raised to a power is equal to that power times the logarithm of the expression itself. The square root of \(x\) can be expressed as \(x^{0.5}\), and 64 as \(4^3\). So, we rewrite \( \log_{4}{\sqrt{x}} - \log_{4}{64} \) as \( 0.5 \times \log_{4}{x} - 3 \).

Step 3: Evaluate Logarithms

Because we're using base 4 logarithm, \(\log_{4}{x}\) cannot be readily evaluated and must be left as is. However, \(\log_{4}{4^3}\) is 3, which we've already factored into our equation in the previous step. Thus, the expanded form of the given logarithmic expression is \(0.5 \times \log_{4}{x} - 3\).

Key Concepts

Logarithm Expansion

Logarithms allow us to express complex relationships in simpler terms by using its expansion properties. When we talk about **logarithm expansion**, what we really mean is breaking down a logarithmic expression into its basic components. This is crucial for solving complex logarithmic equations and for ease of calculation.
One reason to expand logarithms is to simplify expressions for easier evaluation or for solving equations. For instance, given a ratio or product inside a log, the expansion uses known rules like the product and quotient rules to simplify it into smaller log parts. This step not only aids in understanding but also makes further calculations more straightforward by isolating individual components.

Quotient Rule

The quotient rule is a fundamental part of simplifying logarithmic expressions. It states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator.
For example, when faced with an expression like \(\log_{b}\left(\frac{m}{n}\right)\), you can use the quotient rule to rewrite it as \(\log_{b}{m} - \log_{b}{n}\). This allows you to transform a division inside a log into subtraction outside, making it more manageable.

• Helps in breaking down complex ratios into simpler individual parts.
• Useful in solving logarithmic equations without calculators, because you can easily work with the identities or values you already know.

Understanding and applying the quotient rule efficiently can significantly simplify your work, especially in expressions involving fractions.

Power Rule

The power rule simplifies expressions with the exponent within logs. It states proudly: if you have a logarithm with an exponent, you can move that exponent in front as a multiplier.
For example, \(\log_{b}(a^k)\) can be rewritten as \(k \cdot \log_{b}(a)\). This means any power within the log can be simplified by moving it to the front, drastically reducing its complexity.

• Transforms complicated expressions like \(\sqrt{x}\) (which is \(x^{0.5}\)) into a simpler form \(0.5 \times \log_{b}(x)\).
• Makes it easier to multiply the log expression, as you deal with standard multiplication rather than navigating through powers.

The power rule is exceptionally powerful in simplifying and manipulating logarithmic expressions; thus it's a core algebraic tool.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions refers to finding their true numerical value. Some logs can be evaluated directly by understanding their base. For example, \(\log_{4}{64}\) can directly yield a solution if 64 is a power of 4.
However, not all logs are straightforward. If you cannot evaluate expression immediately, focusing on a breakdown or leaving it unchanged might be necessary. For instance, the log base might not be familiar or simplifiable, such as \(\log_{4}(x)\) without any further substitution or simplification.

• Always check if the value inside the log can be expressed as a power of the base to evaluate directly.
• Not all expressions are evaluable, and in such cases, representing them in their expanded form is enough.

Successful evaluation depends on expanding and applying properties meticulously. It demystifies complicated logarithmic data into numbers or leaves it as an understandable expression for further work.