Problem 46 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{b} \sqrt{\frac{3}{y}} $$

Step-by-Step Solution

Answer

\( \log_b \sqrt{\frac{3}{y}} = \frac{1}{2} \log_b 3 - \frac{1}{2} \log_b y \)

Step 1: Recognize the Expression

The given expression is \( \log_b \sqrt{\frac{3}{y}} \). It needs to be rewritten using the properties of logarithms.

Step 2: Convert to Exponent Form

The square root can be rewritten as a fractional exponent: \( \sqrt{\frac{3}{y}} = \left(\frac{3}{y}\right)^{1/2} \).

Step 3: Use Logarithm Power Property

Apply the power rule of logarithms, which states \( \log_b (x^n) = n \cdot \log_b (x) \). Therefore, \( \log_b \left(\frac{3}{y}\right)^{1/2} = \frac{1}{2}\log_b \left(\frac{3}{y}\right) \).

Step 4: Use the Quotient Property of Logarithms

Apply the quotient rule of logarithms, which states \( \log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B \). Thus, \( \log_b \left(\frac{3}{y}\right) = \log_b 3 - \log_b y \).

Step 5: Combine Steps 3 and 4

Substitute the expression obtained in Step 4 into Step 3's formula: \( \frac{1}{2} (\log_b 3 - \log_b y) = \frac{1}{2} \log_b 3 - \frac{1}{2} \log_b y \).

Key Concepts

Logarithm Power Rule

The logarithm power rule is a fundamental property that makes it easier to handle exponents within logarithms. This rule states that for any positive number, the logarithm of a power can be expressed as the exponent multiplied by the logarithm of the base. Mathematically, you can write it as: \[\log_b (x^n) = n \cdot \log_b (x) \]This means if you have a logarithm of an expression raised to a power, that power can "come down" in front of the logarithm as a coefficient.
For example, in the expression \(\log_b \left(\frac{3}{y}\right)^{1/2}\), the exponent \(1/2\) can be brought in front of the logarithm, resulting in: \[\frac{1}{2} \cdot \log_b \left(\frac{3}{y}\right)\]By applying the power rule, complex expressions become more manageable and clearer to interpret. This rule simplifies many logarithmic expressions and is highly useful when dealing with roots and fractional exponents.

Logarithm Quotient Rule

The logarithm quotient rule is another essential property that helps simplify expressions involving division inside a logarithm. It allows you to split a logarithmic expression into a difference of two logarithms. The rule is stated as:\[\log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B\]This means if you have a logarithm of a division, you can express it as the subtraction of two separate logarithms.
For instance, consider applying this to \(\log_b \left(\frac{3}{y}\right)\). Using the quotient rule, you can rewrite it as: \[\log_b 3 - \log_b y\]This transformation into a subtraction form often helps to further simplify expressions, especially when dealing with complex algebraic expressions. The quotient rule is particularly helpful in problems where simplifying or expanding logarithmic expressions is necessary.

Fractional Exponents

Fractional exponents are a way to express roots as exponents, which can be a very useful technique in algebra and calculus. Instead of writing a root symbol, you represent roots using powers of fractions. This allows you to apply the same rules of exponents to roots. If you have:\[\sqrt{x} = x^{1/2}\]Or, more generally, a root of any order can be expressed as:\[x^{1/n} = \sqrt[n]{x}\]In the original problem, the square root \(\sqrt{\frac{3}{y}}\) was expressed as \(\left(\frac{3}{y}\right)^{1/2}\). This conversion makes application of the logarithm power rule straightforward. Fractional exponents are a powerful tool for simplifying expressions, making them more versatile for manipulation using logarithmic properties.

Step-by-Step Solution

To solve problems using logarithms, following a step-by-step approach ensures accuracy and helps grasp each part of the process. Here’s how it applies to the conversion of the expression \(\log_b \sqrt{\frac{3}{y}}\):

• **Step 1:** Identify the expression and plans for simplification: \(\log_b \sqrt{\frac{3}{y}}\).
• **Step 2:** Express the square root with a fractional exponent: \(\left(\frac{3}{y}\right)^{1/2}\).
• **Step 3:** Apply the power rule: \(\frac{1}{2}\log_b \left(\frac{3}{y}\right)\).
• **Step 4:** Use the quotient rule to further simplify: \(\log_b 3 - \log_b y\).
• **Step 5:** Combine these results: \(\frac{1}{2}(\log_b 3 - \log_b y) = \frac{1}{2} \log_b 3 - \frac{1}{2} \log_b y\).

This methodical approach not only makes handling logarithmic expressions straightforward but also builds a strong foundation for tackling more complex problems in mathematics. Following these steps ensures clarity and confidence in solving logarithmic problems.