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. log₆ (36/(√(x+1))

Verified step by step guidance

1. Identify the logarithmic expression: \( \log_6 \left( \frac{36}{\sqrt{x+1}} \right) \). Our goal is to expand this using properties of logarithms.
2. Apply the quotient rule of logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). So, rewrite the expression as \( \log_6 36 - \log_6 \sqrt{x+1} \).
3. Recognize that \( \sqrt{x+1} = (x+1)^{1/2} \). Use the power rule of logarithms: \( \log_b (M^p) = p \log_b M \). So, \( \log_6 \sqrt{x+1} = \frac{1}{2} \log_6 (x+1) \).
4. Substitute back to get the expanded form: \( \log_6 36 - \frac{1}{2} \log_6 (x+1) \).
5. Evaluate \( \log_6 36 \) if possible by expressing 36 as a power of 6. Since \( 36 = 6^2 \), \( \log_6 36 = 2 \). So the fully expanded expression is \( 2 - \frac{1}{2} \log_6 (x+1) \).

Key Concepts

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^p) = p * log_b(M). These rules help break down complex expressions into simpler parts.

Simplifying Radicals Inside Logarithms

When a radical expression appears inside a logarithm, it can be rewritten using fractional exponents. For instance, √(x+1) is equivalent to (x+1)^(1/2). This allows the use of the power rule of logarithms to simplify the expression further.

Evaluating Logarithms Without a Calculator

Some logarithmic expressions can be evaluated exactly by recognizing perfect powers or simplifying the argument. For example, log_6(36) can be simplified since 36 = 6^2, so log_6(36) = 2. This avoids the need for a calculator and provides exact values.