Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ${\log }_4\left(\frac{\sqrt{x}}{64}\right)$

Verified step by step guidance

1. Recall the logarithmic property for quotients: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Apply this to rewrite \(\log_4 \left( \frac{\sqrt{x}}{64} \right)\) as \(\log_4 (\sqrt{x}) - \log_4 (64)\).
2. Express the square root as an exponent: \(\sqrt{x} = x^{\frac{1}{2}}\). Use the power rule of logarithms: \(\log_b (M^p) = p \log_b M\). So, \(\log_4 (\sqrt{x}) = \log_4 (x^{\frac{1}{2}}) = \frac{1}{2} \log_4 x\).
3. Rewrite 64 as a power of 4 if possible. Since \$4^3 = 64\(, we have \)64 = 4^3\(. Use the power rule again: \)\log_4 (64) = \log_4 (4^3) = 3 \log_4 4$.
4. Recall that \(\log_b b = 1\) for any base \(b\). Therefore, \(\log_4 4 = 1\), so \(\log_4 (64) = 3 \times 1 = 3\).
5. Combine all parts to write the expanded form: \(\log_4 \left( \frac{\sqrt{x}}{64} \right) = \frac{1}{2} \log_4 x - 3\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^k) = k * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.

Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of x can be written as x^(1/2). This conversion allows the use of logarithm power rules to simplify expressions involving roots. Recognizing this helps in expanding logarithmic expressions involving radicals.

Change of Base and Evaluating Logarithms

Evaluating logarithms without a calculator often involves expressing numbers as powers of the base. For example, 64 can be written as 4^3 since 4^3 = 64. This allows direct simplification of logarithmic terms like log_4(64) = 3. This concept helps in simplifying and evaluating logarithmic expressions exactly.