Problem 24 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{\sqrt{x}}{25}\right) $$

Step-by-Step Solution

Answer

The fully expanded form of the logarithmic expression is \(1/2 \log _{5} (x) - 2\).

Step 1: Identify the rules

We have a log with a fraction inside the parentheses which means we can apply the quotient rule. We can separate the fraction as follows: \(\log _{5}\left(\frac{\sqrt{x}}{25}\right) = \log _{5}(\sqrt{x}) - \log _{5} (25)\)

Step 2: Evaluate the Right Term

The right term we can simply evaluate without a calculator, since 5^2 equals 25. So \(\log _{5} (25) = 2\)

Step 3: Apply power rule on Left Term

The left term \(\log _{5}(\sqrt{x})\) can be written using the power rule. Since the square root is the same as the power of 1/2, this expression becomes: \(1/2 \log _{5} (x)\)

Step 4: Combine Results

Combine the results from steps 2 and 3. So \(\log _{5}\left(\frac{\sqrt{x}}{25}\right) = 1/2 \log _{5} (x) - 2\)

Key Concepts

Quotient Rule of Logarithms

The quotient rule is a fundamental property of logarithms that helps us simplify expressions that involve division inside a logarithm. Suppose you have a logarithmic function like this: \[ \log_b \left( \frac{M}{N} \right) \]. The quotient rule allows you to express this as: \[ \log_b (M) - \log_b (N) \].
This transformation helps in breaking down complex logarithmic expressions to more manageable parts, making it easier to solve.

• When you see a fraction inside a log, think "quotient rule." It separates the numerator and denominator.
• This rule applies broadly, whether you are working with numbers, variables, or more complicated expressions.

To master this rule, practice it with different logarithmic expressions. Remember that the structure is always a simple subtraction after applying the rule.

Power Rule of Logarithms

The power rule is another vital property when working with logarithms. It gives you the power to "bring down" exponents outside of the log, making calculations simpler. If you have a log expression of the form: \[ \log_b (M^p) \], you can express it as \[ p \times \log_b (M) \].
This rule is especially useful when dealing with powers or roots.

• This rule allows us to simplify expressions involving powers, making it easier to evaluate or manipulate them.
• In the example given, \( \log_5(\sqrt{x}) \) is transformed using the power rule to \( \frac{1}{2} \log_5(x) \).

To utilize this rule effectively, watch for any powers or roots when dealing with logarithms. Recognize that roots are just fractional exponentials.

Evaluating Logarithms

Evaluating logarithms involves simplifying them to the extent possible, often utilizing known values or simplifying expressions. For instance, if you need to evaluate \( \log_b(N) \)and you know that \( b^p = N \), then the evaluation is straightforward: \( p \).

• This process was demonstrated when evaluating \( \log_5(25) \), since \( 5^2 = 25\), leading directly to the solution \( 2 \).
• It's essential to recognize well-known base and power combinations, such as powers of 2, 10, and 5, as these can help quickly evaluate logs without a calculator.

Effective evaluation often combines understanding of these properties with practice, enabling quick and accurate solutions.