Let log_b 2 = A and log_b 3 = C and Write each expression in terms of A and C. log_b √(2/27)

Verified step by step guidance

1. Start by expressing the argument inside the logarithm in terms of powers: \( \sqrt{\frac{2}{27}} = \left( \frac{2}{27} \right)^{\frac{1}{2}} \).
2. Rewrite the fraction inside the parentheses as a quotient of powers: \( \left( \frac{2}{3^3} \right)^{\frac{1}{2}} \) since \(27 = 3^3\).
3. Apply the logarithm power rule: \( \log_b \left( \frac{2}{3^3} \right)^{\frac{1}{2}} = \frac{1}{2} \log_b \left( \frac{2}{3^3} \right) \).
4. Use the logarithm quotient rule: \( \log_b \left( \frac{2}{3^3} \right) = \log_b 2 - \log_b 3^3 \).
5. Apply the logarithm power rule again: \( \log_b 3^3 = 3 \log_b 3 \). Now substitute \( \log_b 2 = A \) and \( \log_b 3 = C \) to write the expression entirely in terms of \( A \) and \( C \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties such as the product, quotient, and power rules. These allow us to rewrite complex logarithmic expressions as sums, differences, or multiples of simpler logarithms, which is essential for expressing the given log in terms of A and C.

Change of Base and Given Logarithm Values

The problem provides logb 2 = A and logb 3 = C, which means logarithms of 2 and 3 with base b are known. Using these values, we can express other logarithms involving 2 and 3 by breaking down numbers into prime factors and substituting A and C accordingly.

Simplifying Radicals and Fractions Inside Logarithms

The expression involves a square root and a fraction inside the logarithm. Understanding how to rewrite radicals as fractional exponents and how to separate logarithms of quotients helps in breaking down the expression into parts that can be expressed using A and C.